Publications (84)
Macroscopic Simulation of Isotropic Permanent Magnets
Florian Bruckner, Claas Abert, Christoph Vogler +6
Accurate simulations of isotropic permanent magnets require to take the magnetization process into account and consider the anisotropic, nonlinear, and hysteretic material behaviou…
Superior bit error rate and jitter due to improved switching field distribution in exchange spring magnetic recording
Dieter Suess, Markus Fuger, Claas Abert +5
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…
Vortex motion in reconfigurable three-dimensional superconducting nanoarchitectures
Elina Zhakina, Luke Turnbull, Weijie Xu +9
When materials are patterned in three dimensions, there exist opportunities to tailor and create functionalities associated with an increase in complexity, the breaking of symmetri…
Laser-induced real-space topology control of spin wave resonances
Tim Titze, Sabri Koraltan, Timo Schmidt +8
Femtosecond laser excitation of materials that exhibit magnetic spin textures promises advanced magnetic control via the generation of ultrafast and non-equilibrium spin dynamics.…
Efficient solution strategy to couple micromagnetic simulations with ballistic transport in magnetic tunnel junctions
Peter Flauger, Claas Abert, Dieter Suess
We present a computationally efficient strategy that allows to simulate magnetization switching driven by spin-transfer torque in magnetic tunnel junctions within a micromagnetic m…
In-situ alignment of anisotropic hard magnets of 3D printed magnets
Maximilian Suppan, Christian Huber, Klaus Mathauer +9
Within this work, we demonstrate in-situ easy-axis alignment of single-crystal magnetic particles inside a polymer matrix using fused filament fabrication. Two different magnetic m…
Energy landscape of noncollinear exchange coupled magnetic multilayers
George Lertzman-Lepofsky, Afan Terko, Sabri Koraltan +3
We conduct an exploration of the energy landscape of two coupled ferromagnetic layers with perpendicular-to-plane uniaxial anisotropy using finite-element micromagnetic simulations…
Sculpting Spin-Wave Landscapes via Curvature of 2D Magnonic Crystals
OndÅej Wojewoda, Robert Kraft, Olha Bezsmertna +11
Engineering the dispersion relation is one of the key ingredients enabling the application of spin waves in computational elements. One way to engineer the spin-wave band structure…
Landau-Lifshitz-Bloch equation for exchange coupled grains
Christoph Vogler, Claas Abert, Florian Bruckner +1
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 fu…
Field- and damping-like spin-transfer torque in magnetic multilayers
Claas Abert, Hossein Sepehri-Amin, Florian Bruckner +3
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 layer…
Solving Large-Scale Inverse Magnetostatic Problems using the Adjoint Method
Florian Bruckner, Claas Abert, Gregor Wautischer +4
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…
Magnetization alignment in spin-transfer-torque magnetic random-access memory
Afan Terko, George Lertzman-Lepofsky, Dieter Suess +2
Reliable operation of perpendicular spin-transfer-torque magnetic random-access memory (p-STT-MRAM) requires control of magnetic alignment within the synthetic antiferromagnet (SAF…
Field-free switching of perpendicular magnetic elements by using two orthogonal sub nanosecond spin orbit torque pulses
Dieter Suess, Claas Abert, Sebastian Zeilinger +2
We propose a field-free switching mechanism that utilizes two spatially orthogonal spin-orbit torque (SOT) currents. Initially applied simultaneously, one of the currents is subseq…
Stochastic ferrimagnetic Landau-Lifshitz-Bloch equation for finite magnetic structures
Christoph Vogler, Claas Abert, Florian Bruckner +1
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-Lif…
Topologically Protected Vortex Structures to Realize Low-Noise Magnetic Sensors
Dieter Suess, Anton Bachleitner-Hofmann, Armin Satz +11
Micromagnetic sensors play a major role towards the miniaturization in the industrial society. The adoption of new and emerging sensor technologies like anisotropic magnetoresistan…
Experimental realisation of a universal inverse-design magnonic device
Noura Zenbaa, Claas Abert, Fabian Majcen +7
In the field of magnonics, which uses magnons, the quanta of spin waves, for energy-efficient data processing, significant progress has been made leveraging the capabilities of the…
Damping Enhancement in YIG at Millikelvin Temperatures due to GGG Substrate
Rostyslav O. Serha, Andrey A. Voronov, David Schmoll +12
Quantum magnonics aims to exploit the quantum mechanical properties of magnons for nanoscale quantum information technologies. Ferrimagnetic yttrium iron garnet (YIG), which offers…
Realization of inverse-design magnonic logic gates
Noura Zenbaa, Fabian Majcen, Claas Abert +6
Magnonic logic gates represent a crucial step toward realizing fully magnonic data processing systems without reliance on conventional electronic or photonic elements. Recently, a…
Single device offset-free magnetic field sensing principle with tunable sensitivity and linear range based on spin-orbit-torques
Sabri Koraltan, Christin Schmitt, Florian Bruckner +12
We propose a novel device concept using spin-orbit-torques to realize a magnetic field sensor, where we eliminate the sensor offset using a differential measurement concept. We der…
On the origin of noncollinear magnetization coupling across RuX layers
Claas Abert, Sabri Koraltan, Florian Bruckner +5
We present a simple atomistic model for the description of noncollinear coupling in magnetic multilayers with hybrid spacer layers made of Ru alloyed to ferromagnetic atoms such as…
GPU Accelerated Atomistic Energy Barrier Calculations of Skyrmion Annihilations
Paul Heistracher, Claas Abert, Florian Bruckner +2
We present GPU accelerated simulations to calculate the annihilation energy of magnetic skyrmions in an atomistic spin model considering dipole-dipole, exchange, uniaxial-anisotrop…
Topology Optimized and 3D Printed Polymer Bonded Permanent Magnets for a Predefined External Field
Christian Huber, Claas Abert, Florian Bruckner +4
Topology optimization offers great opportunities to design permanent magnetic systems that have specific external field characteristics. Additive manufacturing of polymer bonded ma…
Efficient Energy-minimization in Finite-Difference Micromagnetics: Speeding up Hysteresis Computations
Claas Abert, Gregor Wautischer, Florian Bruckner +2
We implement an efficient energy-minimization algorithm for finite-difference micromagnetics that proofs especially useful for the computation of hysteresis loops. Compared to resu…
Micromagnetics and spintronics: Models and numerical methods
Claas Abert
Computational micromagnetics has become an indispensable tool for the theoretical investigation of magnetic structures. Classical micromagnetics has been successfully applied to a…
Efficiently reducing transition curvature in heat-assisted magnetic recording with state-of-the-art write heads
Christoph Vogler, Claas Abert, Florian Bruckner +1
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…
Full analytical solution for the magnetic field of uniformly magnetized cylinder tiles
Florian Slanovc, Michael Ortner, Mohssen Moridi +2
We present an analytical solution for the magnetic field of a homogeneously magnetized cylinder tile and by extension solutions for full cylinders, rings, cylinder sectors and ring…
Self-consistent solution of magnetic and friction energy losses of a magnetic nanoparticle
Santiago Helbig, Claas Abert, Pedro A. Sánchez +2
We present a simple simulation model for analysing magnetic and frictional losses of magnetic nano-particles in viscous fluids subject to alternating magnetic fields. Assuming a pa…
Tailoring the energy landscape of a Bloch point singularity with curvature
Sandra Ruiz-Gomez, Claas Abert, Pamela Morales-Fernández +14
Topological defects, or singularities, play a key role in the statics and dynamics of complex systems. In magnetism, Bloch point singularities represent point defects that mediate…
magnum.np -- A PyTorch based GPU enhanced Finite Difference Micromagnetic Simulation Framework for High Level Development and Inverse Design
Florian Bruckner, Sabri Koraltan, Claas Abert +1
magnum.np is a micromagnetic finite-difference library completely based on the tensor library PyTorch. The use of such a high level library leads to a highly maintainable and exten…
Reconfigurable three dimensional magnetic nanoarchitectures
Sabri Koraltan, Fabrizio Porrati, Robert Kraft +6
Three-dimensional (3D) nanomagnetism is a rapidly developing field within magnetic materials research, where exploiting the third dimension unlocks opportunities for innovative app…
Highly Parallel Demagnetization Field Calculation Using the Fast Multipole Method on Tetrahedral Meshes with Continuous Sources
Pietro Palmesi, Lukas Exl, Florian Bruckner +2
The long-range magnetic field is the most time-consuming part in micromagnetic simulations. Improvements both on a numerical and computational basis can relief problems related to…
FFT-based Kronecker product approximation to micromagnetic long-range interactions
Lukas Exl, Claas Abert, Norbert J. Mauser +3
We derive a Kronecker product approximation for the micromagnetic long range interactions in a collocation framework by means of separable sinc quadrature. Evaluation of this opera…
Exchange spin-wave propagation in Ga:YIG nanowaveguides
Andrey A. Voronov, Khrystyna O. Levchenko, Roman Verba +7
Spin-wave-based computing has emerged as a promising approach to overcome the fundamental limitations of CMOS technologies. However, the increasing demand for device miniaturizatio…
Comparison of Sensitivity and Low Frequency Noise Contributions in GMR and TMR Spin Valve Sensors with a Vortex State Free Layer
Herbert Weitensfelder, Hubert Brueckl, Armin Satz +9
Magnetoresistive spin valve sensors based on the giant- (GMR) and tunnelling- (TMR) magnetoresisitve effect with a flux-closed vortex state free layer design are compared by means…
3D Printing of Polymer Bonded Rare-Earth Magnets With a Variable Magnetic Compound Density for a Predefined Stray Field
Christian Huber, Claas Abert, Florian Bruckner +7
Additive manufacturing of polymer bonded magnets is a recently developed technique, for single-unit production, and for structures that have been impossible to manufacture previous…
AC noise reduction based on exchange coupled grains for heat-assisted-magnetic recording: The effect of an FeRh interlayer
Christoph Vogler, Claas Abert, Florian Bruckner +1
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. Reco…
Chiral switching and dynamic barrier reductions in artificial square ice
Naëmi Leo, Matteo Pancaldi, Sabri Koraltan +10
Collective dynamics in lithographically-defined artificial spin ices offer profound insights into emergent correlations and phase transitions of geometrically-frustrated Ising spin…
k-Selective Electrical-to-Magnon Transduction with Realistic Field-distributed Nanoantennas
Andreas Höfinger, Andrey A. Voronov, David Schmoll +9
The excitation and detection of propagating spin waves with lithographed nanoantennas underpin both classical magnonic circuits and emerging quantum technologies. Here, we establis…
A self-consistent spin-diffusion model for micromagnetics
Claas Abert, Michele Ruggeri, Florian Bruckner +4
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…
Magnetization reversal mechanism of double-helix nanowires probed by dark-field magneto-optical Kerr effect
Takeaki Gokita, Jakub Jurczyk, Naëmi Leo +8
Double-helix (DH) nanowires provide a platform to study the influence of geometric chirality on spin chirality. Their three-dimensional (3D) helical architecture and tunable inter-…
Magnetic anisotropy and GGG substrate stray field in YIG films down to millikelvin temperatures
Rostyslav O. Serha, Andrey A. Voronov, David Schmoll +17
Quantum magnonics investigates the quantum-mechanical properties of magnons such as quantum coherence or entanglement for solid-state quantum information technologies at the nanosc…
Proposal for a micromagnetic standard problem: domain wall pinning at phase boundaries
Paul Heistracher, Claas Abert, Florian Bruckner +2
We propose a novel micromagnetic standard problem calculating the coercive field for unpinning a domain wall at the interface of a multiphase magnet. This problem is sensitive to d…
A repulsive skyrmion chain as guiding track for a race track memory
Dieter Suess, Christoph Vogler, Florian Bruckner +1
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…
Spin-polarized transport in ferromagnetic multilayers: An unconditionally convergent FEM integrator
Claas Abert, Gino Hrkac, Marcus Page +3
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. Th…
A Fourier-Space Approach to Physics-Informed Magnetization Reconstruction from Nitrogen-Vacancy Measurements
Alexander Setescak, Florian Bruckner, Dieter Suess +8
Reconstructing magnetization textures from nitrogen-vacancy (NV) magnetometry stray-field measurements is a challenging, fundamentally ill-posed inverse problem, further complicate…
Generation and annihilation of skyrmions and antiskyrmions in magnetic heterostructures
Sabri Koraltan, Claas Abert, Florian Bruckner +3
We demonstrate the controlled generation and annihilation of (anti)skyrmions with tunable chirality in magnetic heterostructures by means of micromagnetic simulations. By making us…
Large Scale Finite-Element Simulation of Micromagnetic Thermal Noise
Florian Bruckner, Massimiliano d'Aquino, Claudio Serpico +3
An efficient method for the calculation of ferromagnetic resonant modes of magnetic structures is presented. Finite-element discretization allows flexible geometries and location d…
Areal density optimizations for heat-assisted-magnetic recording of high density bit-patterned media
Christoph Vogler, Claas Abert, Florian Bruckner +2
Heat-assisted-magnetic recording (HAMR) is hoped to be the future recording technique for high density storage devices. Nevertheless, there exist several realizations strategies. W…
Transition from ferromagnetic to noncollinear to paramagnetic state with increasing Ru concentration in FeRu films
Juliana Lisik, Manuel Rojas, Spencer Myrtle +11
The structural and magnetic properties of sputter-deposited FeRu films were studied for . The crystal structure of FeRu is shown to be predom…
Steerable current-driven emission of spin waves in magnetic vortex pairs
Sabri Koraltan, Katrin Schultheiss, Florian Bruckner +4
The efficient excitation of spin waves is a key challenge in the realization of magnonic devices. We demonstrate the current-driven generation of spin waves in antiferromagneticall…
Wavenumber-dependent magnetic losses in YIG-GGG heterostructures at millikelvin temperatures
David Schmoll, Andrey A. Voronov, Rostslav O. Serha +7
Magnons have inspired potential applications in modern quantum technologies and hybrid quantum systems due to their intrinsic nonlinearity, nanoscale scalability, and a unique set…
Dipolar-stabilized first and second-order antiskyrmions in ferrimagnetic multilayers
Michael Heigl, Sabri Koraltan, Marek VaÅatka +13
Skyrmions and antiskyrmions are topologically protected spin structures with opposite topological charge. Particularly in coexisting phases, these two types of magnetic quasi-parti…
Fundamental limits in heat assisted magnetic recording and methods to overcome it with exchange spring structures
Dieter Suess, Christoph Vogler, Claas Abert +3
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 f…
Accurate finite-difference micromagnetics of magnets including RKKY interaction -- analytical solution and comparison to standard micromagnetic codes
Dieter Suess, Sabri Koraltan, Florian Slanovc +2
Within this paper we show the importance of accurate implementations of the RKKY interactions for antiferromagnetically coupled ferromagnetic layers with thicknesses exceeding the…
Parallel-in-Time Integration of the Landau-Lifshitz-Gilbert Equation with the Parallel Full Approximation Scheme in Space and Time
Robert Kraft, Sabri Koraltan, Markus Gattringer +3
Speeding up computationally expensive problems, such as numerical simulations of large micromagnetic systems, requires efficient use of parallel computing infrastructures. While pa…
Significant reduction of critical currents in MRAM designs using dual free layer with perpendicular and in-plane anisotropy
Dieter Suess, Christoph Vogler, Florian Bruckner +2
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 report…
Considering non-uniform current distributions in magnetoresistive sensor designs and their implications for the resistance transfer function
Anton Bachleitner-Hofmann, Claas Abert, Hubert Brückl +5
Non-uniform current distributions of spin valves with disk shaped free layers are investigated. In the context of spin valves, the vortex state, which is the ground-state in many d…
Heat assisted magnetic recording of bit patterned media beyond 10 Tb/in
Christoph Vogler, Claas Abert, Florian Bruckner +2
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 presen…
Basic noise mechanisms of heat-assisted-magnetic recording
Christoph Vogler, Claas Abert, Florian Bruckner +2
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 t…
Spin Torque Efficiency and Analytic Error Rate Estimates of Skyrmion Racetrack Memory
Dieter Suess, Christoph Vogler, Florian Bruckner +2
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 para…
Skyrmionic device for three dimensional magnetic field sensing enabled by spin-orbit torques
Sabri Koraltan, Rahul Gupta, Reshma Peremadathil Pradeep +14
Magnetic skyrmions are topologically protected local magnetic solitons that are promising for storage, logic or general computing applications. In this work, we demonstrate that we…
The 2026 Skyrmionics Roadmap
Sabri Koraltan, Claas Abert, Manfred Albrecht +64
Magnetic skyrmions and related topological spin textures have emerged as a central topic in condensed-matter physics, combining fundamental significance with potential for transfor…
Micromagnetic Simulation and Optimization of Spin-Wave Transducers
Florian Bruckner, Kristýna DavÃdková, Claas Abert +2
The increasing demand for higher data volume and faster transmission in modern wireless telecommunication systems has elevated requirements for 5G high-band RF hardware. Spin-Wave…
Dependence of energy barrier reduction on collective excitations in square artificial spin ice: A comprehensive comparison of simulation techniques
Sabri Koraltan, Matteo Pancaldi, Naëmi Leo +9
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 squa…
A three-dimensional spin-diffusion model for micromagnetics
Claas Abert, Michele Ruggeri, Florian Bruckner +4
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…
Modeling Magnetoelastic Wave Interactions in Magnetic Films and Heterostructures: A finite-difference approach
Peter Flauger, Matthias KüÃ, Michael Karl Steinbauer +8
The (inverse) magnetostrictive effect in ferromagnets couples the magnetic properties to the mechanical stress, allowing for an interaction between the magnetic and mechanical degr…
Inverse-design topology optimization of magnonic devices using level-set method
Andrey A. Voronov, Marcos Cuervo Santos, Florian Bruckner +3
The inverse design approach in magnonics exploits the wave nature of magnons and machine learning to develop logical devices with functionalities that exceed the capabilities of an…
Magnetically Programmable Surface Acoustic Wave Filters: Device Concept and Predictive Modeling
Michael K. Steinbauer, Peter Flauger, Matthias Küà +7
Filtering surface acoustic wave (SAW) signals of specified frequencies depending on the strength of an external magnetic field in a magnetostrictive material has garnered significa…
magnum.fe: A micromagnetic finite-element simulation code based on FEniCS
Claas Abert, Lukas Exl, Florian Bruckner +2
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 th…
Generation of Bloch Points with Controlled Spin Texture Using Geometrical Boundary Conditions
Naëmi Leo, Daniel Wolf, Alicia Estela Herguedas Alonso +16
Bloch points are three-dimensional topological singularities in magnetization that play a key role in topological transformations of spin textures, such as skyrmion creation or ann…
A fast finite-difference algorithm for topology optimization of permanent magnets
Claas Abert, Christian Huber, Florian Bruckner +3
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 e…
Contactless and absolute linear displacement detection based upon 3D printed magnets combined with passive radio-frequency identification
Roman Windl, Claas Abert, Florian Bruckner +4
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 magne…
jaxFMM: An Adaptive, GPU-Parallel Implementation of the Fast Multipole Method in JAX
Robert Kraft, Florian Bruckner, Dieter Suess +1
The paper introduces jaxFMM, an open-source, adaptive implementation of the Fast Multipole Method for the Laplace kernel built with JAX, which exploits JIT compilation and GPU para…
Perspectives on inverse design for AI magnonics
Franz Vilsmeier, Florian Bruckner, Claas Abert +2
Inverse design - specifying a desired functionality and letting a computational algorithm find the optimal structure - has emerged as a powerful paradigm for magnonic device engine…
Non-planar geometrical effects on the magnetoelectrical signal in a three-dimensional nanomagnetic circuit
Fanfan Meng, Claire Donnelly, Claas Abert +11
Expanding nanomagnetism and spintronics into three dimensions (3D) offers great opportunities for both fundamental and technological studies. However, probing the influence of comp…
A full-fledged micromagnetic code in less than 70 lines of NumPy
Claas Abert, Florian Bruckner, Christoph Vogler +3
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 finit…
Hybrid FFT algorithm for fast demagnetization field calculations on non-equidistant magnetic layers
Paul Heistracher, Florian Bruckner, Claas Abert +2
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 pre…
Influence of grain size and exchange interaction on the LLB modeling procedure
Christoph Vogler, Claas Abert, Florian Bruckner +2
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…
Thermal Stoner-Wohlfarth Model for Magnetodynamics of Single Domain Nanoparticles: Implementation and Validation
Deniz Mostarac, Andrey A. Kuznetsov, Santiago Helbig +4
We present the thermal Stoner-Wohlfarth (tSW) model and apply it in the context of Molecular Dynamics simulations. The model is validated against an ensemble of immobilized, random…
NeuralMag: an open-source nodal finite-difference code for inverse micromagnetics
Claas Abert, Florian Bruckner, Andrey Voronov +11
We present NeuralMag, a flexible and high-performance open-source Python library for micromagnetic simulations. NeuralMag leverages modern machine learning frameworks, such as PyTo…
Interlinking helical spin textures in nanopatterned chiral magnets
Luke Alexander Turnbull, Max Thomas Birch, Marisel Di Pietro MartÃnez +15
Nanoscale topologically non-trivial magnetization configurations generate significant interest due to both the fundamental properties of their knotted structures and their potentia…
Numerical Methods for the Stray-Field Calculation: A Comparison of recently developed Algorithms
Claas Abert, Lukas Exl, Gunnar Selke +2
Different numerical approaches for the stray-field calculation in the context of micromagnetic simulations are investigated. We compare finite difference based fast Fourier transfo…
Tension-free Dirac strings and steered magnetic charges in 3D artificial spin ice
Sabri Koraltan, Florian Slanovc, Florian Bruckner +5
3D nano-architectures present a new paradigm in modern condensed matter physics with numerous applications in photonics, biomedicine, and spintronics. They are promising for the re…
Back hopping in spin-transfer-torque devices, possible origin and counter measures
Claas Abert, Hossein Sepehri-Amin, Florian Bruckner +3
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 p…