Magnetoelastic Energy Research Articles

The Interplay of Core Diameter and Diameter Ratio on the Magnetic Properties of Bistable Glass-Coated Microwires

Glass-coated microwires exhibiting magnetic bistability have garnered significant attention as promising wireless sensing elements, primarily due to their rapid magnetization switching capabilities. These microwires consist of a metallic core with diameter d, encased in a glass coating, with a total diameter D. In this study, we investigated how the dimensions of both components and their ratio (d/D) influence the magnetization reversal behavior of Fe-based microwires. While previous studies have focused on either d or d/D individually, our research uniquely considered the combined effect of both parameters to provide a comprehensive understanding of their impact on magnetic properties. The metallic core diameter d varied from 10 to 19 µm and the d/D ratio was in the range of 0.48–0.68. To assess the magnetic properties of these microwires, including the shape of the hysteresis loop, coercivity, remanent magnetization, and the critical length of bistability, we employed vibrating sample magnetometry in conjunction with FORC-analysis. Additionally, to determine the critical length of bistability, magnetic measurements were conducted on microwires with various lengths, ranging from 1.5 cm down to 0.05 cm. Our findings reveal that coercivity is primarily dependent on the d/D parameter. These observations are effectively explained through an analysis that considers the competition between magnetostatic and magnetoelastic anisotropy energies. This comprehensive study paves the way for the tailored design of glass-coated microwires for diverse wireless sensing applications.

Exploiting a novel magnetoelastic tunable bi-stable energy converter for vibration energy mitigation

Exploiting a novel magnetoelastic tunable bi-stable energy converter for vibration energy mitigation

Energy and Structure of the Terbium Domain Wall

The domain wall energy is calculated by the balance between exchange, magnetocrystalline anisotropy and magnetoelastic energy contributions. The described method is theoretical and is based on experimental measurements of neutron inelastic scattering. The domain wall energy is determined by both finding the minimum of energy and deriving the energy and setting it to zero. The determination was undertaken for the discrete case, and this means that the calculation was performed for each plane or atomic layer. This is in contrast with the Bloch wall, which assumes continuum mean. The energy of the Lilley domain wall was discussed. Most of the energy of the Bloch wall was comprised inside the Lilley distance (above 99.9% of the energy). Antiferromagnetic interactions strongly decreased the domain wall energy. The negative terms due to antiferromagnetism must be considered in the Hamiltonian describing the exchange energy terms. The domain wall energy and width of terbium were reassessed. The values varied between 83.7 and 95.2 Kelvin (10.3 to 11.2 ergs/cm2). The domain width was estimated to be 57 Angstroms. It was found that a significant part of the total domain wall energy was concentrated on the planes at the center of the domain wall.

Hard-magnetic soft magnetoelastic materials: Energy considerations

In the finite magnetoelastic deformation context, the Zeeman magnetic energy on its own is found to be inconsistent with the general theory of nonlinear magnetoelasticity. An appropriate additional term, depending on the definition of the reference configuration magnetization, is used to modify the total magnetoelastic energy to confer consistency. In particular, this confirms that the total Cauchy stress tensor is symmetric, in contrast to recent conclusions in the literature where the Zeeman magnetic energy was utilized.

Thermodynamic description for magneto-plastic coupling in electrical steel sheets

The purpose of the study is to propose a thermodynamic description of the full magneto-mechanical coupling in electrical steel sheets, including both elasticity and plasticity influence. Kinematic and isotropic hardening are considered as state variables and included to a second-order magneto-elastic energy written as a function of cubic invariants. The simulation of magnetic behaviour of plastified samples subjected to several elastic stresses reproduce the general trends of measurements carried out on non-oriented Fe-3%Si sheets.

Research on the microphysical mechanism of metal magnetic memory testing through 45 # steel under compressive stress

To explore the microphysical mechanism of metal magnetic memory testing (MMMT) technique, the effect of compressive stress on the domain structure in 45# steel specimen and the pinning effect of stress-induced defects on domain walls (DWs) were investigated by in-situ Lorentz Transmission Electron Microscopy (LTEM). The changes in domain structure and defects are observed simultaneously. And the deformation and magnetic moment distribution of the specimen under compressive stress were measured and analyzed. The results show that, the rotation of magnetic moment corresponding to the deformation of the specimen can greatly reduce the magnetoelastic energy according to the calculation of magnetoelastic energy. When stress is removed, the domain structures do not restore, and DWs’ pinning effect of defects could be clearly observed. The results provide a vigorous support for the memory effect of ferromagnetic material on historical stress.

Magnetoelastic Losses in the Linear Response Region for Three- and Four-Axis Idealized Magnets

In the presented studies, a description of the losses related to relaxation processes in idealized three- and four-axis ferromagnets, as well as ferrites, is considered for the case of a linear response (small perturbations) on the basis of a macroscopic approach. The description takes into account reversible 90° displacements of domain walls (DW) of concentrations (с), ways of DW pinning by linear defects, and processes of DW displacements and rotations of the spontaneous magnetization vector I⃗s, caused by a longitudinal elastic stress wave σ. As a result, the anisotropy of amplitude-independent losses emerges due to displacement processes. Frequency dependence of losses is obtained with consideration of the magnetic symmetry of the crystal, its magnetostructural parameters, and the concentration of magnetic phases ci. The paper also describes the internal frictionn Q-1, the absorption coefficient a, the velocity of propagation of elastic vibrations v and the Δ (1/E)-effect, thus allowing to reveal the texture in the distribution of DWs by their types.
 It is shown that if the relaxation times of rotation processes τ = β / 2K1 and displacement processes τc = βс / mω20 are the same, then the maxima of the dependence of internal friction on frequency Qc-1(ω) and QB-1(ω) superimpose on each other.
 The revealed features of magneto-elastic energy dissipation are of interest for laboratory studies and have promising applications in practice for determining the orientation of crystals, describing the texture, and calculating the differential ΔЕ-effect in magnets. The features of the ΔЕ-effect for the considered cases are also revealed.

Phonon-induced magnetization dynamics in Co-doped iron garnets

The developing field of strain-induced magnetization dynamics offers a promising path toward efficiently controlling spins and phase transitions. Understanding the underlying mechanisms is crucial in finding the optimal parameters supporting the phononic switching of magnetization. Here, we present an experimental and numerical study of time-resolved magnetization dynamics driven by the resonant excitation of an optical phonon mode in iron garnets. Upon pumping the latter with an infrared pulse obtained from a free-electron laser, we observe spatially varying magnetization precession, with its phase depending on the direction of an external magnetic field. Our micromagnetic simulations effectively describe the magnetization precession and switching in terms of laser-induced changes in the crystal's magneto-elastic energy.

Generation of imprinted strain gradients for spintronics

In this work, we propose and evaluate an inexpensive and CMOS-compatible method to locally apply strain on a Si/SiOx substrate. Due to high growth temperatures and different thermal expansion coefficients, a SiN passivation layer exerts a compressive stress when deposited on a commercial silicon wafer. Removing selected areas of the passivation layer alters the strain on the micrometer range, leading to changes in the local magnetic anisotropy of a magnetic material through magnetoelastic interactions. Using Kerr microscopy, we experimentally demonstrate how the magnetoelastic energy landscape, created by a pair of openings, enables in a magnetic nanowire the creation of pinning sites for in-plane vortex walls that propagate in a magnetic racetrack. We report substantial pinning fields up to 15 mT for device-relevant ferromagnetic materials with positive magnetostriction. We support our experimental results with finite element simulations for the induced strain, micromagnetic simulations, and 1D model calculations using the realistic strain profile to identify the depinning mechanism. All the observations above are due to the magnetoelastic energy contribution in the system, which creates local energy minima for the domain wall at the desired location. By controlling domain walls with strain, we realize the prototype of a true power-on magnetic sensor that can measure discrete magnetic fields or Oersted currents. This utilizes a technology that does not require piezoelectric substrates or high-resolution lithography, thus enabling wafer-level production.

A mechanics model of hard-magnetic soft rod with deformable cross-section under three-dimensional large deformation

Hard-magnetic soft (HMS) materials are soft active materials prepared by embedding the hard-magnetic particles (e.g. NdFeB) into soft elastomer, which can be actuated by applied magnetic fields. With the ability to retain remnant magnetism, HMS materials have many applications in soft robotics, haptic sensors and other fields. For HMS structures used in soft robotics, mechanics models have been developed for better exploiting the potential of the HMS materials. However, for rod-like structures made of soft materials, the nonlinear stress–strain relation and the areal change of the cross-section have not been considered in previous modeling work. In the present three-dimensional HMS rod model, started with the geometrically exact beam theory, the rigid cross-section assumption is replaced with a planar areal change assumption for soft rods, incompressible neo-Hookean material model is used to model the hyperelastic behavior of elastomers, which will be more accurate than linear relation under large deformation. Consistent with the deformable cross-section assumption, the magneto-elastic energy distribution is formulated and reduced to the line of centroid of the soft rod. To verify the accuracy and efficiency of the model, some simulation examples and experiments are performed. The error between the experimental results and the simulation results calculated by using 5 elements is within 10%. Our model can accurately and efficiently predict the 3D large deformation of the HMS rod, which can reduce the design and optimization cost and shorten the design cycle of HMS structure.

Suppression of skyrmion Hall effect via standing surface acoustic waves in hybrid ferroelectric/ferromagnetic heterostructures

Magnetic skyrmion is a promising information carrier for its low critical driven current density, topological stability, and small size, which has been proposed for various devices such as racetrack memory and logic gates. However, the skyrmion Hall effect originating from Magnus force leads to transverse motion, which hinders the development of skyrmionic device applications. Here, we propose artificial tracks built by standing surface acoustic waves (SSAWs) to suppress the skyrmion Hall effect through micromagnetic simulations. We systematically study the dynamics of an isolated skyrmion under SSAWs and driven currents in a prototype of the ferromagnetic skyrmion system. The skyrmion Hall angle changes from 80° to 0°, where the skyrmion motion is along the driven current. An analytical model considering magnetoelastic energy induced by SSAWs is developed, and a linear relation between the current density and the critical SSAW amplitude to eliminate the skyrmion Hall effect is achieved. Furthermore, a reconfigurable multichannel skyrmion racetrack is constructed through the change of SSAW wavelengths. Our work opens a feasible route for the suppression of skyrmion Hall effect via SSAWs.

Conduction mechanism, dielectric, and magnetic investigation of lithium ferrite thin films deposited by pulsed laser deposition

The demand for ferrites in the form of thin films enhanced in the last few years due to the growth in microwave and semiconductor industries. LiDy0.1Fe4.9O8 films and their capacitors are fabricated on Pt(111)/Ti/SiO2/Si having different thicknesses using PLD. The formation of phase are confirmed by XRD and Raman spectra. The lattice constant and strain decreased, whereas the average grain size increased with the increase in thickness. The XPS spectra confirmed the presence of all the constituent elements with an appropriate valance state. The surface micrograph evident the inhibition of uniform grain growth having minimal roughness up to 13 nm. The dielectric constant increased with the thickness and temperature and reduced with the frequency, whereas the loss tangent decreased with the film thickness. The in-plane and out-of-plane magnetization degraded, whereas the coercivity enhanced with the film thickness. This variation is explained based on the lattice mismatch, grain size, and magnetoelastic energy density. Out of the various conduction mechanisms, Mott's VRH is recognized as the most appropriate mechanism for the deposited films. The varying thickness of a film is an effective parameter for tuning the physical properties of the film. The observed results suggest that LDFO films are promising for magnetic oxide semiconductor applications.

Modeling of the morphic effect using a vanishing 2nd order magneto-elastic energy

The study takes place in the context of magneto-elastic coupling. A new magnetoelastic energy formulation is proposed at the magnetic domain scale in order to account for the non-linear and non-monotonous effect of stress on magnetization and magnetostriction. The model uses 1st order cubic invariants and a vanishing 2nd order cubic invariants.

A Nonlinear Magnetoelastic Energy Model and Its Application in Domain Wall Velocity Prediction.

In this letter, we propose a nonlinear Magnetoelastic Energy (ME) with a material parameter related to electron interactions. An attenuating term is contained in the formula of the proposed nonlinear ME, which can predict the variation in the anisotropic magneto-crystalline constants induced by external stress more accurately than the classical linear ME. The domain wall velocity under stress and magnetic field can be predicted accurately based on the nonlinear ME. The proposed nonlinear ME model is concise and easy to use. It is important in sensor analysis and production, magneto-acoustic coupling motivation, magnetoelastic excitation, etc.

Structural and magnetic asymmetry at the interfaces of MgO/FeCoB/MgO trilayer: Precise study under x-ray standing wave conditions

Magnetic tunnel junctions based on FeCoB as a magnetic electrode and MgO as a tunneling barrier gained much attention because of their applications in random access memories and magnetic sensors in disk drives. In this work, the structural and magnetic properties of the MgO/FeCoB/MgO trilayer have been studied precisely under x-ray standing wave (XSW) conditions, where XSW is generated through a high-density (Pt) waveguide structure. The combined x-ray scattering and fluorescence data obtained under XSW conditions revealed the formation of a high-density FeCoB layer at the MgO/FeCoB interface (FeCoB-on-MgO) in the as-deposited trilayer. Diffusion of B from the FeCoB layer into MgO is attributed to the formation of Fe- and Co-rich high-density layer (B-deficient FeCoB layer) at the interface. Angular-dependent magnetism of the trilayer structure revealed the presence of in-plane magnetic anisotropy (IMA), which disappeared with thermal annealing at a temperature of 450 °C. Stress in B-deficient FeCoB layer at the interface is attributed to the origin of IMA through magneto-elastic anisotropy energy minimization. The disappearance of anisotropy after annealing is mainly due to the removal of long-range stress and the formation of crystalline bcc-FeCo phase.

Magnetic domain structure of epitaxial Gd films grown on W(110)

We present a detailed real-space spin-polarized scanning tunneling microscopy (SP-STM) study of the magnetic domain structure of Gd(0001) films epitaxially grown on W(110). To find optimal preparation conditions, the influence of the substrate temperature during deposition and of the postgrowth annealing temperature was investigated. Our results show that the lowest density of surface defects, such as step edges as well as screw and edge dislocations, is obtained for room-temperature deposition and subsequent annealing at 900 K. SP-STM data reveal small-size magnetic domains at lower annealing temperatures, evidently caused by pinning at grain boundaries and other crystalline defects. The coverage-dependent magnetic domain structure of optimally prepared Gd films was systematically investigated. For low coverage up to about 80 atomic layers (AL), we observe $\ensuremath{\mu}\mathrm{m}$-size domains separated by domain walls which are oriented approximately along the $[1\overline{1}0]$ direction of the underlying W substrate. Above a critical film thickness ${\mathrm{\ensuremath{\Theta}}}_{\text{crit}}\ensuremath{\approx}(100\ifmmode\pm\else\textpm\fi{}20)$ AL, we identify stripe domains, indicative of a spin reorientation transition from in plane to out of plane. In agreement with existing models, the periodicity of the stripe domains increases the further the coverage exceeds ${\mathrm{\ensuremath{\Theta}}}_{\text{crit}}$. While the orientation of the stripe domains is homogeneous over large distances just above ${\mathrm{\ensuremath{\Theta}}}_{\text{crit}}$, we find a characteristic zigzag pattern at $\mathrm{\ensuremath{\Theta}}\ensuremath{\gtrsim}200$ AL and irregular stripe domains beyond 500 AL. Intermediate minima and maxima of the magnetic signal indicate the nucleation of branching domains. The results are discussed in terms of various contributions to the total magnetic energy, such as the magnetocrystalline, magnetostatic, and magnetoelastic energy density.

A geometrically exact model for thin magneto-elastic shells

We develop a reduced model for hard-magnetic, thin, linear-elastic shells that can be actuated through an external magnetic field, with geometrically exact strain measures. Assuming a reduced kinematics based on the Kirchhoff–Love assumption, we derive a reduced two-dimensional magneto-elastic energy that can be minimized through numerical analysis. In parallel, we simplify the reduced energy by expanding it up to the second order in the displacement field and provide a physical interpretation. Our theoretical analysis allows us to identify and interpret the two primary mechanisms dictating the magneto-elastic response: a combination of equivalent magnetic pressure and forces at the first order, and distributed magnetic torques at the second order. We contrast our reduced framework against a three-dimensional nonlinear model by investigating three test cases involving the indentation and the pressure buckling of shells under magnetic loading. We find excellent agreement between the two approaches, thereby verifying our reduced model for shells undergoing nonlinear and non-axisymmetric deformations. We believe that our model for magneto-elastic shells will serve as a valuable tool for the rational design of magnetic structures, enriching the set of reduced magnetic models.

On Calculation of the Stiffness Coefficients of Domain Boundaries in Magnetic and Electric Ordered Systems in Linear and Weakly Nonlinear Domains

This paper presents the study of the features of domain boundaries dynamics based on the macroscopic approach. In addition, a methodology for the calculation of domain boundary stiffness coefficient due to its importance among the relaxation parameters of magnetic and electric ordered systems is included in the study.
 The proposed algorithm allows the calculation of stiffness coefficients of domain boundaries (DB) in ferromagnetic, ferroelectric, and ferroelectromagnetic materials in linear and weakly nonlinear DB displacement areas caused by an external force. The external force is identified as the difference in the densities of magnetoelastic (electroelastic) energies of crystals for domains separated by the DB. It is related to the difference of their energies because each domain has its own orientation of spontaneous magnetization IS and polarization PS vectors at the same equal values of external stress components σij. The desired stiffness coefficients of the DB are calculated as the second derivatives of magnetoelastic (electroelastic) subsystem energies with respect to DB displacements. The stiffness coefficients of the first k1 and the second k2 order of smallness are obtained in a similar way using cubic and biquadratic components of energies in terms of DB displacements, i.e. relative deformation of crystals.

Quasi-static strain governing ultrafast spin dynamics

The quasi-static strain (QSS) is the product induced by the lattice thermal expansion after ultrafast photo-excitation. Although the ultrafast spin dynamics driven by the QSS and thermal effects are barely distinguishable in time, they should be treated separately because of their different fundamental actions. By employing ultrafast Sagnac interferometry and the magneto-optical Kerr effect, we demonstrate quantitatively the existence of QSS and the decoupling of two effects counteracting each other in typical polycrystalline Co and Ni films. The Landau-Lifshitz-Gilbert and Kittel equations considering a magnetoelastic energy term showed that QSS, rather than the thermal energy, in ferromagnets plays a governing role in ultrafast spin dynamics. This demonstration provides a way to analyze ultrafast photo-induced phenomena.

A theory of magneto-elastic nanorods obtained through rigorous dimension reduction

Starting from a two-dimensional theory of magneto-elasticity for fiber-reinforced magnetic elastomers we carry out a rigorous dimension reduction to derive a rod model that describes a thin magneto-elastic strip undergoing planar deformations. The main features of the theory are the following: a magneto-elastic interaction energy that manifests itself through a distributed torque; a penalization term that prevents local interpenetration of matter; a regularization term that depends on the second gradient of the deformation and models microstructure-induced size effects. As an application, we consider a problem involving magnetically-induced buckling and we study how the intensity of the field at the onset of the instability increases if the length of the rod is decreased. Finally, we assess the accuracy of the deduced model by performing numerical simulations where we compare the two-dimensional and the one-dimensional theories in some special cases, and we observe excellent agreement.