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Collective dynamics in lithographically-defined artificial spin ices offer profound insights into emergent correlations and phase transitions of geometrically-frustrated Ising spin systems. Their temporal and spatial evolution are often simulated using kinetic Monte Carlo simulations, which rely on the precise knowledge of the switching barriers to obtain predictive results in agreement with experimental observations. In many cases, however, the barriers are derived from simplified assumptions only, and do not take into account the full physical picture of nanomagnetic switching. Here we describe how the immediate magnetic environment of a nanomagnet reversing via quasi-coherent rotation can induce clockwise and counter-clockwise switching channels with different barrier energies. This barrier splitting for chiral reversal channels can be sizeable and, as string-method micromagnetic simulations show, is relevant for artificial spin ice systems made of both exchange -- as well as magnetostatically --dominated units. Due to the barrier splitting (and further reductions due to non-uniform reversal) transition rates can be exponentially enhanced by several orders of magnitude compared to mean-field predictions, especially in the limit of rare switching events where thermal excitation is less likely. This leads to significantly faster relaxation time scales and modified spatial correlations. Our findings are thus of integral importance to achieve realistic kinetic Monte Carlo simulations of emergent correlations in artificial spin systems, magnonic crystals, or the evolution of nanomagnetic logic circuits., Comment: 14 pages, 9 figures

Skyrmions and antiskyrmions are topologically protected spin structures with opposite topological charge. Particularly in coexisting phases, these two types of magnetic quasi-particles may show fascinating physics and potential for spintronic devices. While skyrmions are observed in a wide range of materials, until now antiskyrmions were exclusive to materials with D2d symmetry. In this work, we show first and second-order antiskyrmions stabilized by magnetic dipole-dipole interaction in Fe/Gd-based multilayers. We modify the magnetic properties of the multilayers by Ir insertion layers. Using Lorentz transmission electron microscopy imaging, we observe coexisting antiskyrmions, Bloch skyrmions, and type-2 bubbles and determine the range of material properties and magnetic fields where the different spin objects form and dissipate. We perform micromagnetic simulations to obtain more insight into the studied system and conclude that the reduction of saturation magnetization and uniaxial anisotropy leads to the existence of this zoo of different spin objects and that they are primarily stabilized by dipolar interaction.

Magnetoresistive (xMR) sensors find extensive application in science and industry, replacing Hall sensors in various low field environments. While there have been some efforts in increasing the dynamic field range of xMR sensors, Hall sensors remain to dominate high field applications due to their wide linear range. Using a perpendicular magnetized reference system and an in-plane free layer allows us to overcome this disadvantage of xMR sensors, and, furthermore, investigate spin-canting effects in interlayer exchange coupled perpendicular synthetic antiferromagnets (p-SAF). We created p-SAFs with exchange coupling fields of up to 10 kOe, based on magnetic Co/Pt multilayer systems. The p-SAFs are either designed as "single" p-SAFs, where two Co/Pt multilayers are interlayer exchange coupled via a 4 {\AA} thick Ru spacer, or as "double" p-SAFs, where an additional Co layer is interlayer exchange coupled to the top multilayer. These p-SAFs are used for giant magnetoresistance (GMR) sensors with wide dynamic field range. By using a p-SAF as the reference system and employing an in-plane magnetic layer as the GMR's free layer, the linear range can be effectively increased limited only by the p-SAF's switching fields. Additionally, the magnetic anisotropy of the in-plane free layer is fully controlled, which allows saturation fields by design. Different configurations were investigated, ranging from free layer magnetic saturation at lower to far higher fields than the p-SAF's switching fields. We can show through micromagnetic simulations that certain GMR transfer curves are dominated by spin-canting effects in the interlayer exchange coupled reference system. Finally, our simulation results lay out the correlation of the p-SAF's design parameters and its magnetization reversal behavior., Comment: 9 pages, 10 figures, 1 table

In this study, the magnetic reversal process of exchange-coupled bilayer systems, consisting of a ferrimagnetic TbFeCo alloy layer and a ferromagnetic [Co/Ni/Pt]N multilayer, was investigated. In particular, minor loop studies, probing solely the reversal characteristics of the softer ferromagnetic layer, reveal two distinct reversal mechanisms, which depend strongly on the thickness of the ferromagnetic layer. For thick layers, irreversible switching of the macroscopic minor loop is observed. The underlying microscopic origin of this reversal process was studied in detail by high-resolution magnetic force microscopy, showing that the reversal is triggered by in-plane domain walls propagating through the ferromagnetic layer. In contrast, thin ferromagnetic layers show a hysteresis-free reversal, which is nucleation-dominated due to grain-to-grain variations in magnetic anisotropy of the Co/Ni/Pt multilayer and an inhomogeneous exchange coupling with the magnetically hard TbFeCo layer, as confirmed by micromagnetic simulations., Comment: 8 pages, 7 figures

We perform micromagnetic simulations to study the switching barriers in square artificial spin ice systems consisting of elongated single domain magnetic islands arranged on a square lattice. By considering a double vertex composed of one central island and six nearest neighbor islands, we calculate the energy barriers between two types of double vertices by applying the string method. We investigate by means of micromagnetic simulations the consequences of the neighboring islands, the inhomogeneities in the magnetization of the islands and the reversal mechanisms on the energy barrier by comparing three different approaches with increasing complexity. The micromagnetic models, where the string method is applied, are compared to the currently common method, the mean barrier approximation. Our investigations indicate that a proper micromagnetic modeling of the switching process leads to significantly lower energy barriers, by up to 35% compared to the mean-barrier approximation, so decreasing the expected average life time up to seven orders of magnitude. Hereby, we investigate the influence of parallel switching channels and the conceptional approach of using a mean-barrier to calculate the corresponding rates., Comment: 21 pages, 7 figures

Exchange-coupled structures consisting of ferromagnetic and ferrimagnetic layers become technologically more and more important. We show experimentally the occurrence of completely reversible, hysteresis-free minor loops of [Co(0.2 nm)/Ni(0.4 nm)/Pt(0.6 nm)]$_N$ multilayers exchange-coupled to a 20 nm thick ferrimagnetic Tb$_{28}$Co$_{14}$Fe$_{58}$ layer, acting as hard magnetic pinning layer. Furthermore, we present detailed theoretical investigations by means of micromagnetic simulations and most important a purely analytical derivation for the condition of the occurrence of full reversibility in magnetization reversal. Hysteresis-free loops always occur if a domain wall is formed during the reversal of the ferromagnetic layer and generates an intrinsic hard-axis bias field that overcomes the magnetic anisotropy field of the ferromagnetic layer. The derived condition further reveals that the magnetic anisotropy and the bulk exchange of both layers, as well as the exchange coupling strength and the thickness of the ferromagnetic layer play an important role for its reversibility., Comment: 9 pages, 7 figures

The comprehensive simulation of magnetic recording, including the write and read-back process, on granular media becomes computationally expensive if the magnetization dynamics of each grain are explicitly computed. In addition, in heat-assisted magnetic recording, the writing of a single track becomes a random process since the temperature must be considered and thermal noise is involved. Further, varying grain structures of various granular media must also be taken into account to obtain correct statistics for the final read-back signal. Hence, it requires many repetitions of the write process to investigate the mean signal as well as the noise. This work presents a method that improves the statistical evaluation of the whole recording process. The idea is to avoid writing the magnetization to one of its binary states. Instead, we assign each grain its probability of occupying one of its stable states, which can be calculated in advance in terms of a switching probability phase diagram. In the read-back process, we combine the probabilities to calculate a mean signal and its variance. Afterwards, repetitions on different media lead to the final read-back signal. Using a recording example, we show that the statistical behavior of the evaluated signal-to-noise ratio can be significantly improved by applying this probability mapping method, while the computational effort remains low., Comment: 5 pages, 4 figures, 2019 MMM Conference

In micromagnetic simulations, the demagnetization field is by far the computationally most expensive field component and often a limiting factor in large multilayer systems. We present an exact method to calculate the demagnetization field of magnetic layers with arbitrary thicknesses. In this approach we combine the widely used fast-Fourier-transform based circular convolution method with an explicit convolution using a generalized form of the Newell formulas. We implement the method both for central processors and graphics processors and find that significant speedups for irregular multilayer geometries can be achieved. Using this method we optimize the geometry of a magnetic random-access memory cell by varying a single specific layer thickness and simulate a hysteresis curve to determine the resulting switching field.

The reduction of the transition curvature of written bits in heat-assisted magnetic recording (HAMR) is expected to play an important role for the future areal density increase of hard disk drives. Recently a write head design with flipped write and return poles was proposed. In this design a large spatial field gradient of the write head was the key to significantly reduce the transition curvature. In this work we optimized the write pole of a heat-assisted magnetic recording head in order to produce large field gradients as well as large fields in the region of the heat pulse. This is done by topology optimization. The simulations are performed with dolfin-adjoint. For the maximum field gradients of $8.1\,$mT/nm, $8.6\,$mT/nm and $11.8\,$mT/nm, locally resolved footprints of an FePt like hard magnetic recording medium are computed with a coarse-grained Landau-Lifshitz-Bloch (LLB) model and the resulting transition curvature is analysed. Additional simulations with a bilayer structure with $50\%$ hard and $50\%$ soft magnetic material are computed. The results show that for both recording media, the optimized head design does not lead to any significant improvement of the written track. Thus, we analyse the transition curvature for the optimized write heads theoretically with an effective recording time window (ERTW) model. Moreover, we check how higher field gradients influence the curvature reduction. The results show that a simple optimization of the conventional head design design is not sufficient for effective curvature reduction. Instead, new head concepts will be needed to reduce transition curvature.

We optimize the recording medium for heat-assisted magnetic recording by using a high/low $T_{\mathrm{c}}$ bilayer structure to reduce AC and DC noise. Compared to a former work, small Gilbert damping $\alpha=0.02$ is considered for the FePt like hard magnetic material. Atomistic simulations are performed for a cylindrical recording grain with diameter $d=5\,$nm and height $h=8\,$nm. Different soft magnetic material compositions are tested and the amount of hard and soft magnetic material is optimized. The results show that for a soft magnetic material with $\alpha_{\mathrm{SM}}=0.1$ and $J_{ij,\mathrm{SM}}=7.72\times 10^{-21}\,$J/link a composition with $50\%$ hard and $50\%$ soft magnetic material leads to the best results. Additionally, we analyse how much the areal density can be improved by using the optimized bilayer structure compared to the pure hard magnetic recording material. It turns out that the optimized bilayer design allows an areal density that is $1\,$Tb/in$^2$ higher than that of the pure hard magnetic material while obtaining the same SNR., Comment: 7 pages, 7 figures

In magnetic recording the signal-to-noise ratio (SNR) is a good indicator for the quality of written bits. However, a priori it is not clear which parameters have the strongest influence on the SNR. In this work, we investigate the role of the Gilbert damping on the SNR. Grains consisting of FePt like hard magnetic material with two different grain sizes $d_1=5\,$nm and $d_2=7\,$nm are considered and simulations of heat-assisted magnetic recording (HAMR) are performed with the atomistic simulation program VAMPIRE. The simulations display that the SNR saturates for damping constants larger or equal than 0.1. Additionally, we can show that the Gilbert damping together with the bit length have a major effect on the SNR whereas other write head and material parameters only have a minor relevance on the SNR., Comment: 7 pages, 4 figures

The signal-to-noise ratio (SNR) of a bit series written with heat-assisted magnetic recording (HAMR) on granular media depends on a large number of different parameters. The choice of material properties is essential for the obtained switching probabilities of single grains and therefore for the written bits' quality in terms of SNR. Studies where the effects of different material compositions on transition jitter and the switching probability are evaluated were done, but it is not obvious, how significant those improvements will finally change the received SNR. To investigate that influence, we developed an analytical model of the switching probability phase diagram, which contains independent parameters for, inter alia, transition width, switching probability and curvature. Different values lead to corresponding bit patterns on granular media, where a reader model detects the resulting signal, which is finally converted to a parameter dependent SNR value. For grain diameters between 4 and 8nm, we show an increase of ~10dB for bit lengths between 4 and 12nm, an increase of ~9dB for maximum switching probabilities between 0.64 and 1.00, a decrease of ~5dB for down-track-jitter parameters between 0 and 4nm and an increase of ~1dB for reduced bit curvature. Those results are furthermore compared to the theoretical formulas for the SNR. We obtain a good agreement, even though we show slight deviations., Comment: 9 pages, 18 Figures

An efficient method for the calculation of ferromagnetic resonant modes of magnetic structures is presented. Finite-element discretization allows flexible geometries and location dependent material parameters. The resonant modes can be used for a semi-analytical calculation of the power spectral density of the thermal white-noise, which is relevant for many sensor applications. The proposed method is validated by comparing the noise spectrum of a nano-disk with time-domain simulations.

We investigate how a temperature reduction in z-direction influences the switching probability and the noise in heat-assisted magnetic recording (HAMR) for a bit in bit-patterned media with dimensions d=5nm and h=10nm. Pure hard magnetic bits are considered and simulations with a continuous laser pulse are performed using the atomistic simulation tool VAMPIRE. The results display that the switching behavior shows a thermally induced exchange spring effect. Simultaneously, both the AC and the DC noise increase. Additionally, we illustrate how an artificial Curie temperature gradient within the material can compensate the HAMR performance loss due to the temperature gradient. Further, due to the graded Curie temperature, DC noise can be reduced compared to a structure where no temperature gradient is considered., Comment: arXiv admin note: text overlap with arXiv:1709.03764

We present GPU accelerated simulations to calculate the annihilation energy of magnetic skyrmions in an atomistic spin model considering dipole-dipole, exchange, uniaxial-anisotropy and Dzyaloshinskii-Moriya interactions using the simplified string method. The skyrmion annihilation energy is directly related to its thermal stability and is a key measure for the applicability of magnetic skyrmions to storage and logic devices. We investigate annihilations mediated by Bloch points as well as annihilations via boundaries for various interaction energies. Both processes show similar behaviour, with boundary annihilations resulting in slightly smaller energy barriers than Bloch point annihilations., Comment: Corrected sign in dipole term (equation 1 and 7) and adapted the phase diagram in figure 3. Updated acknowledgement

Precise modeling of the magnetization dynamics of nanoparticles with finite size effects at fast varying temperatures is a computationally challenging task. Based on the Landau-Lifshitz-Bloch (LLB) equation we derive a coarse grained model for disordered ferrimagnets, which is both fast and accurate. First, we incorporate stochastic fluctuations to the existing ferrimagnetic LLB equation. Further, we derive a thermodynamic expression for the temperature dependent susceptibilities, which is essential to model finite size effects. Together with the zero field equilibrium magnetization the susceptibilities are used in the stochastic ferrimagnetic LLB to simulate a $5\times10$ nm$^2$ ferrimagnetic GdFeCo particle with 70 % FeCo and 30 % Gd under various external applied fields and heat pulses. The obtained trajectories agree well with those of an atomistic model, which solves the stochastic Landau-Lifshitz-Gilbert equation for each atom. Additionally, we derive an expression for the intergrain exchange field which couple the ferromagnetic sublattices of a ferrimagnet. A comparison of the magnetization dynamics obtained from this simpler model with those of the ferrimagnetic LLB equation shows a perfect agreement.

Micromagnetic sensors play a major role towards the miniaturization in the industrial society. The adoption of new and emerging sensor technologies like anisotropic magnetoresistance (AMR), giant magnetoresistance (GMR) and tunnel magnetoresistance (TMR) sensors are mainly driven by their integrability and enhanced sensitivity. At the core of such sensors, a microstructured ferromagnetic thin film element transduces the magnetic signal. Such elements usually switch via multi-domain, C- or S-shaped magnetization states and, therefore, often exhibit an open non-linear hysteresis curve. Linearity and hysteretic effects, as well as magnetic noise are key features in the improvement of such sensors. Here, we report on the physical origin of these disturbing factors and the inherent connection of noise and hysteresis. Critical noise sources are identified by means of analytic and micromagnetic models. The dominant noise source is due to irreproducible magnetic switching of the transducer element at external fields close to the Stoner Wohlfarth switching field. Furthermore, a solution is presented to overcome these limiting factors: a disruptive sensor design is proposed and analyzed which realizes a topologically protected magnetic vortex state in the transducer element. Compared to state of the art sensors the proposed sensor layout has negligible hysteresis, a linear regime about an order of magnitude higher and lower magnetic noise making the sensor ideal candidate for applications ranging from automotive industry to biological application.

Within this work a passive and wireless magnetic sensor, to monitor linear displacements is proposed. We exploit recent advances in 3D printing and fabricate a polymer bonded magnet with a spatially linear magnetic field component corresponding to the length of the magnet. Regulating the magnetic compound fraction during printing allows specific shaping of the magnetic field distribution. A giant magnetoresistance magnetic field sensor is combined with a radio-frequency identification tag in order to passively monitor the exerted magnetic field of the printed magnet. Due to the tailored magnetic field, a displacement of the magnet with respect to the sensor can be detected within the sub-mm regime. The sensor design provides good flexibility by controlling the 3D printing process according to application needs. Absolute displacement detection using low cost components and providing passive operation, long term stability and longevity renders the proposed sensor system ideal for structural health monitoring applications., Comment: 4 pages, 5 figures, 1 table

It is assumed that heat-assisted magnetic recording (HAMR) is the recording technique of the future. For pure hard magnetic grains in high density media with an average diameter of $5$nm and a height of $10$nm the switching probability is not sufficiently high for the use in bit-patterned media. Using a bilayer structure with 50$\%$ hard magnetic material with low Curie temperature and 50$\%$ soft magnetic material with high Curie temperature to obtain more than 99.2$\%$ switching probability, leads to very large jitter. We propose an optimized material composition to reach a switching probability of $P_{\mathrm{switch}}>99.2\%$ and simultaneously achieve the narrow transition jitter of pure hard magnetic material. Simulations with a continuous laser spot were performed with the atomistic simulation program VAMPIRE for a single cylindrical recording grain with a diameter of 5nm and a height of 10nm. Different configurations of soft magnetic material and different amounts of hard and soft magnetic material were tested and discussed. Within our analysis, a composition with 20$\%$ soft magnetic and $80\%$ hard magnetic material reaches the best results with a switching probability $P_{\mathrm{switch}}>99.2\%$, an off-track jitter parameter $\sigma_{\mathrm{off},80/20}=14.2$K and a down-track jitter parameter $\sigma_{\mathrm{down},80/20}=0.49$nm., Comment: 7 pages, 10 figures

We present a finite-difference method for the topology optimization of permanent magnets that is based on the FFT accelerated computation of the stray-field. The presented method employs the density approach for topology optimization and uses an adjoint method for the gradient computation. Comparsion to various state-of-the-art finite-element implementations shows a superior performance and accuracy. Moreover, the presented method is very flexible and easy to implement due to various preexisting FFT stray-field implementations that can be used.

A skyrmion racetrack design is proposed that allows for thermally stable skyrmions to code information and dynamical pinning sites that move with the applied current. This concept solves the problem of intrinsic distributions of pinning times and pinning currents of skyrmions at static geometrical or magnetic pinning sites. The dynamical pinning sites are realized by a skyrmion carrying wire, where the skyrmion repulsion is used in order to keep the skyrmions at equal distances. The information is coded by an additional layer where the presence and absence of a skyrmion is used to code the information. The lowest energy barrier for a data loss is calculated to be DE = 55 kBT300 which is sufficient for long time thermal stability.

High storage density and high data rate are two of the most desired properties of modern hard disk drives. Heat-assisted magnetic recording (HAMR) is believed to achieve both. Recording media, consisting of exchange coupled grains with a high and a low $T_{\mathrm{C}}$ part, were shown to have low DC noise, but increased AC noise, compared to hard magnetic single phase grains, like FePt [1]. In this work we extensively investigate the influence of an FeRh interlayer on the magnetic noise in exchange coupled grains. We find an optimal grain design that reduces the jitter in down-track direction by up to 30 % and in off-track direction by up to 50 %, depending on the head velocity, compared to the same structures without FeRh. Further, the mechanisms causing this jitter reduction are demonstrated. Additionally, we show that for ultrashort heat pulses and low write temperatures the switching time distribution of the analyzed grain structure is reduced by a factor of four, compared to the same structure without FeRh layer. This feature could be interesting for HAMR with a pulsed laser spot and could resume the discussion about this HAMR technique.

In this paper the thermal stability of skyrmion bubbles and the critical currents to move them over pinning sites is investigated. For the used pinning geometries and the used parameters, the unexpected behavior is reported that the energy barrier to overcome the pinning site is larger than the energy barrier of the annihilation of a skyrmion. The annihilation takes place at boundaries by current driven motion as well as due to the excitation over energy barriers, in the absence of currents, without forming Bloch points. It is reported that the pinning sites, which are required to allow thermally stable bits, significantly increase the critical current densities to move the bits in skyrmion like structures to about $j_{crit}$ = 0.62 TA/m$^2$. These currents are similar to those obtained experimentally to move stable skyrmions at room temperature. By calculating the thermal stability as well as the critical current, we can derive the spin torque efficiency $\eta$ = $\Delta/I_c = 0.19 k_BT_{300}~/\mu$A, which is in a similar range to the simulated spin torque efficiency of MRAM structures. Finally, it is shown that the stochastic depinning process of any racetrack like device requires extremely narrow depinning time distribution smaller than ~6% of the current pulse length to reach bit error rates smaller than ${10^{ - 9}}$.

The curvature of bit transitions on granular media is a serious problem for the read-back process. We address this fundamental issue and propose a possibility to efficiently reduce transition curvatures with state-of-the-art heat-assisted magnetic recording (HAMR) heads. We compare footprints of conventional with those of the proposed head design on different media, consisting of exchange coupled and single phase grains. Additionally, we investigate the impact of various recording parameters, like the full width at half maximum (FWHM) of the applied heat pulse and the coercivity gradient near the write temperature of the recording grains. The footprints are calculated with a coarse grained model, based on the Landau-Lifshitz-Bloch (LLB) equation. The presented simulations show a transition curvature reduction of up to 40 %, in the case of a medium with exchange coupled grains and a heat pulse with a FWHM of 40 nm. We further give the reason for the straightening of the bit transitions, by means of basic considerations with regard to the effective recording time window (ERTW) of the write process. Besides the transition curvature reduction the proposed head design yields an improvement of the transition jitter in both down-track and off-track direction.

The effect of undesirable high-frequency free-layer switching in magnetic multilayer systems, referred to as back hopping, is investigated by means of the spin-diffusion model. A possible origin of the back-hopping effect is found to be the destabilization of the pinned layer which leads to perpetual switching of both layers. The influence of different material parameters on the critical switching currents for the free and pinned layer is obtained by micromagnetic simulations. It is found that the choice of a free-layer material with low polarization $\beta$ and saturation magnetization $M_s$, and a pinned-layer material with high $\beta$ and $M_s$ leads to a low free-layer critical current and a high pinned-layer critical current and hence reduces the likelihood of back hopping. While back hopping was observed in various types of devices, there are only few experiments that exhibit this effect in perpendicularly magnetized systems. However, our simulations suggest, that this is likely to change due to loss of pinned-layer anisotropy when decreasing device sizes.

Topology optimization offers great opportunities to design permanent magnetic systems that have specific external field characteristics. Additive manufacturing of polymer bonded magnets with an end-user 3D printer can be used to manufacture permanent magnets with structures that have been difficult or impossible to manufacture previously. This work combines these two powerful methods to design and manufacture permanent magnetic system with specific properties. The topology optimization framework is simple, fast, and accurate. It can be also used for reverse engineering of permanent magnets in order to find the topology from field measurements. Furthermore, a magnetic system that generate a linear external field above the magnet is presented. With a volume constraint the amount of magnetic material can be minimized without losing performance. Simulations and measurements of the printed system show a very good agreement.

One essential feature in MRAM cells is the spin torque efficiency, which describes the ratio of the critical switching current to the energy barrier. Within this paper it is reported that the spin torque efficiency can be improved by a factor of 3.2 by the use a of dual free layer device, which consists of one layer with perpendicular crystalline anisotropy and a second layer with in-plane crystalline anisotropy. Detailed simulations solving the spin transport equations simultaneously with the micromagnetics equation were performed in order to understand the origin of the switching current reduction by a factor of 4 for the dual layer structure compared to a single layer structure. The main reason could be attributed to an increased spin accumulation within the free layer due to the dynamical tilting of the magnetization within the in-plane region of the dual free layer.

Additive manufacturing of polymer bonded magnets is a recently developed technique, for single-unit production, and for structures that have been impossible to manufacture previously. Also new possibilities to create a specific stray field around the magnet are triggered. The current work presents a method to 3D print polymer bonded magnets with a variable magnetic compound density distribution. A low-cost, end-user 3D printer with a mixing extruder is used to mix permanent magnetic filaments with pure PA12 filaments. The magnetic filaments are compounded, extruded, and characterized for the printing process. To deduce the quality of the manufactured magnets with a variable compound density, an inverse stray field framework is used. The effectiveness of the printing process and the simulation method is shown. It can also be used to manufacture magnets that produce a predefined stray field in a given region. Examples for sensor applications are presented. This setup and simulation framework allows the design and manufacturing of polymer bonded permanent magnets which are impossible to create with conventional methods.

We investigate the spin-transfer torque in a magnetic multilayer structure by means of a spin-diffusion model. The torque in the considered system, consisting of two magnetic layers separated by a conducting layer, is caused by a perpendicular-to-plane current. We compute the strength of the field-like and the damping-like torque for different material parameters and geometries. Our studies suggest that the field-like torque highly depends on the exchange coupling strength of the itinerant electrons with the magnetization both in the pinned and the free layer. While a low coupling leads to very high field-like torques, a high coupling leads to low or even negative field-like torques. The dependence of the different torque terms on system parameters is considered very important for the development of applications such as STT MRAM and spin-torque oscillators.

An efficient algorithm for the reconstruction of the magnetization state within magnetic components is presented. The occurring inverse magnetostatic problem is solved by means of an adjoint approach, based on the Fredkin-Koehler method for the solution of the forward problem. Due to the use of hybrid FEM-BEM coupling combined with matrix compression techniques the resulting algorithm is well suited for large-scale problems. Furthermore the reconstruction of the magnetization state within a permanent magnet is demonstrated., Comment: 5 pages, 4 figures

Reliably predicting bit-error rates in realistic heat-assisted magnetic recording simulations is a challenging task. Integrating the Landau-Lifshitz-Bloch (LLB) equation can reduce the computational effort to determine the magnetization dynamics in the vicinity of the Curie temperature. If one aims that these dynamics coincide with trajectories calculated from the atomistic Landau-Lifshitz-Gilbert equation, one has to carefully model required temperature dependent material functions such as the zero-field equilibrium magnetization as well as the parallel and normal susceptibilities. We present an extensive study on how these functions depend on grain size and exchange interactions. We show that, if the size or the exchange constant of a reference grain is modified, the material functions can be scaled, according to the changed Curie temperature, yielding negligible errors. This is shown to be valid for volume changes of up to $\pm 40$ % and variations of the exchange constant of up to $\pm10$ %. Besides the temperature dependent material curves, computed switching probabilities also agree well with probabilities separately determined for each system. Our study suggest that there is no need to recalculate the required LLB input functions for each particle. Within the presented limits it is sufficient to scale them to the Curie temperature of the altered system.

Heat-assisted magnetic recording (HAMR) is expected to be a key technology to significantly increase the areal storage density of magnetic recording devices. At high temperatures thermally induced noise becomes a major problem, which must be overcome in order to reliably write magnetic bits with narrow transitions. We propose an elementary model based on the effective recording time window (ERTW) to compute the switching probability of bits during HARM of bit-patterned media. With few assumptions this analytical model allows to gain deeper insights into the basic noise mechanisms like AC and DC noise. Finally, we discuss strategies to reduce noise and to increase the areal storage density of both bit-patterned as well as granular media.

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.

Heat-assisted-magnetic recording (HAMR) is hoped to be the future recording technique for high density storage devices. Nevertheless, there exist several realizations strategies. With a coarse-grained Landau-Lifshitz-Bloch (LLB) model we investigate in detail benefits and disadvantages of continuous and pulsed laser spot recording of shingled and conventional bit-patterned media. Additionally we compare single phase grains and bits having a bilayer structure with graded Curie temperature, consisting of a hard magnetic layer with high $T_{\mathrm{C}}$ and a soft magnetic one with low $T_{\mathrm{C}}$, respectively. To describe the whole write process as realistic as possible a distribution of the grain sizes and Curie temperatures, a displacement jitter of the head and the bit positions are considered. For all these cases we calculate bit error rates of various grain patterns, temperatures and write head positions to optimize the achievable areal storage density. Within our analysis shingled HAMR with a continuous laser pulse moving over the medium reaches the best results, and thus having the highest potential to become the next generation storage device.

We report two effects that lead to a reduction of the switching field distribution in exchange spring media. The first effect relies on a subtle mechanism of the interplay between exchange coupling between soft and hard layers and anisotropy that allows significant reduction of the switching field distribution in exchange spring media. This effect reduces the switching field distribution by about 30% compared to single-phase media. A second effect is that due to the improved thermal stability of exchange spring media over single-phase media, thermal fluctuation leads to reduced fundamental transition jitter. The influence of this overall improved switching field distribution on the transition jitter in granular recording and the bit error rate in bit-patterned magnetic recording is discussed. The transition jitter in granular recording for a distribution of K1 values of 3% in the hard layer, taking into account thermal fluctuations during recording, is estimated to be a = 0.78 nm, which is similar to the best reported calculated jitter in optimized heat-assisted recording media.

The limits of the areal storage density as can be achieved with heat assisted magnetic recording (HAMR) are still an open issue. We want to address this central question and present the design of a possible bit patterned medium with an areal storage density above 10 Tb/in$^2$. The model uses hard magnetic recording grains with 5 nm diameter and 10 nm height. It assumes a realistic distribution of the Curie temperature of the underlying material as well as a realistic distribution of the grain size and the grain position. In order to compute the areal density we analyze the detailed switching behavior of a recording bit under different external conditions, which allows to compute the bit error rate of a recording process (shingled and conventional) for different grain spacings and write head positions. Hence, we are able to optimize the areal density of the presented medium.

Accurate simulations of isotropic permanent magnets require to take the magnetization process into account and consider the anisotropic, nonlinear, and hysteretic material behaviour near the saturation configuration. An efficient method for the solution of the magnetostatic Maxwell equations including the description of isotropic permanent magnets is presented. The algorithm can easily be implemented on top of existing finite element methods and does not require a full characterization of the hysteresis of the magnetic material. Strayfield measurements of an isotropic permanent magnet and simulation results are in good agreement and highlight the importance of a proper description of the isotropic material.

Ensuring a permanent increase of magnetic storage densities is one of the main challenges in magnetic recording. Conventional approaches based on single phase grains are not suitable to achieve this goal, because their grain volume is limited due to the superparamagnetic limit. Grains with graded anisotropy are the most promising candidates to overcome this limit, providing magnetic memory bits with small volumes, low coercivity and high thermal stability at the same time. Combining micromagnetic simulations with forward flux sampling (FFS), a computational method for rare events that has been recently applied to the magnetic nanostructures, we have determined thermal escape rates and attempt frequencies of a graded media grain and two single phase grains of the same geometry. We find that graded anisotropy can increase the thermal stability of a grain by 12 orders of magnitudes from tens of milliseconds to centuries without changing the coercive field.

We present a complete micromagnetic finite-difference code in less than 70 lines of Python. The code makes largely use of the NumPy library and computes the exchange field by finite differences and the demagnetization field with a fast convolution algorithm. Since the magnetization in finite-difference micromagnetics is represented by a multi-dimensional array and the NumPy library features a rich interface for this data structure, the presented code is a good starting point for the development of novel algorithms.

The switching probability of magnetic elements for heat assisted recording is investigated. It is found that FePt elements with a diameter of 5 nm and a height of 10nm show, at a field of 0.5 T, thermally written in errors of 12 percent, which is significant too large for bit patterned magnetic recording. Thermally written in errors can be decreased if larger head fields are applied. However, larger fields lead to an increase the fundamental thermal jitter. This leads to a dilemma between thermally written in errors and fundamental thermal jitter. This dilemma can be partly relaxed by increasing the thickness of the FePt film up to 30nm. For realistic head fields, it is found that the fundamental thermal jitter is in the same order of magnitude of the fundamental thermal jitter in conventional recording, which is about 0.5 to 0.8 nm. Composite structures consisting of high Curie top layer and FePt as hard magnetic storage layer can reduce the thermally written in errors to be smaller than 10-4 if the damping constant is increased in the soft layer. Large damping may be realized by doping with rare earth elements. Similar to single FePt grains also in composite structure an increase of switching probability is sacrifices by an increase of thermal jitter. Structures utilizing first order phase transitions breaking the thermal jitter and writeability dilemma are discussed.

We implement a finite-element scheme that solves the Landau-Lifshitz-Gilbert equation coupled to a diffusion equation accounting for spin-polarized currents. The latter solves for the spin accumulation not only in magnetic materials but also in nonmagnetic conductors. The presented method incorporates the model by Slonczewski for the description of spin torque in magnetic multilayers as well as the model of Zhang and Li for the description of current driven domain-wall motion. Furthermore it is able to do both resolve the time evolution of the spin accumulation or treat it in an adiabatic fashion by the choice of sufficiently large time steps.

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.

We present an analytical derivation of the switching field distribution for a single domain particle from the N\'eel-Brown model in the presence of a linearly swept magnetic field and influenced by thermal fluctuations. We show that the switching field distribution corresponds to a probability density function and can be obtained by solving a master equation for the not-switching probability together with the transition rate for the magnetization according to the Arrhenius-N\'eel Law. By calculating the first and second moments of the probability density function we succeed in modeling rate-dependent coercivity and the standard deviation of the coercive field. Complementary to the analytical approach, we also present a Monte Carlo simulation for the switching of a macrospin, which allows us to account for the field dependence of the attempt frequency. The results show excellent agreement with results from a Langevin dynamics simulation and therefore point out the importance to include the relevant dependencies in the attempt frequency. However, we conclude that the N\'eel-Brown model fails to predict switching fields correctly for common field rates and material parameters used in magnetic recording from the loss of normalization of the probability density function. Investigating the transition regime between thermally assisted and dynamic switching will be of future interest regarding the development of new magnetic recording technologies.