Magnetocrystalline Anisotropy Research Articles

Topological Hall Effect Instigated in Kagome Mn$_{3-x}$Sn due to Mn-deficit Induced Noncoplanar Spin Structure.

Magnetic topological semimetals are manifestations of the interplay between electronic and magnetic phases of matter, leading to peculiar characteristics such as the anomalous Hall effect (AHE) and the topological Hall effect (THE). Mn$_{3}$Sn is a time-reversal symmetry-broken (TRS) magnetic Weyl semimetal showing topological characteristics within the Kagome lattice network. This study reveals a large topological Hall effect in Mn$_{2.8}$Sn (6\% Mn deficit Mn$_3$Sn) at room temperature in the $xy$-plane, despite being an antiferromagnet. We argue that the magnetocrystalline anisotropy induced noncoplanar spin structure is responsible for the observed topological Hall effect in these systems. Further, the topological properties of these systems are highly anisotropic, as we observe a large anomalous Hall effect in the $zx$-plane. We find that Fe doping at the Mn site, Mn$_{3-x}$Fe$_x$Sn ($x$=0.2, 0.25, and 0.35), tunes the topological properties of these systems. These findings promise the realization of potential topotronic applications at room temperature.

Relationship between orbital moment anisotropy, magnetocrystalline anisotropy, and Dzyaloshinskii-Moriya interaction in W/Co/Pt trilayers

We have studied the Co layer thickness dependences of magnetocrystalline anisotropy (MCA), Dzyaloshinskii-Moriya interaction (DMI), and orbital moment anisotropy (OMA) in W/Co/Pt trilayers, in order to clarify their correlations with each other. We find that the MCA favors magnetization along the film normal and monotonically increases with decreasing effective magnetic layer thickness (teff). The magnitude of the Dzyaloshinskii-Moriya exchange constant (|D|) increases with decreasing teff until teff∼1 nm, below which |D| decreases. The MCA and |D| scale with 1/teff for teff larger than ∼1 nm, indicating an interfacial origin. The increase of MCA with decreasing teff continues below teff∼ 1 nm, but with a slower rate. To clarify the cause of the teff dependences of MCA and DMI, the OMA of Co in W/Co/Pt trilayers is studied using x-ray magnetic circular dichroism. We find nonzero OMA when teff is smaller than ∼0.8 nm. The OMA increases with decreasing teff more rapidly than what is expected from the MCA, indicating that factors other than OMA contribute to the MCA at small teff. The teff dependence of the OMA also suggests that |D| at teff smaller than ∼1 nm is not related to the OMA at the interface. We propose that the growth of Co on W results in a strain and/or texture that reduces the interfacial DMI, and, to some extent, MCA at small teff. Published by the American Physical Society 2024

Metallic Ca Aggregates Formed Along Ion Tracks and Optical Anisotropy in CaF2 Crystals Irradiated with Swift Heavy Ions

It is known that swift heavy ion (SHI) irradiation induces the shape elongation of metal nanoparticles (NPs) embedded in transparent insulators, which results in anisotropic optical absorption. Here, we report another type of the optical anisotropy induced in CaF2 crystals without including intentionally embedded metal NPs. The CaF2 samples were irradiated with 200 MeV Xe14+ ions with an incident angle of 45° from the surface normal. With the increasing fluence, an absorption band at ~550 nm, which is ascribed to Ca aggregates, increases both the intensity and the anisotropy. XTEM observation clarified the formation of the continuous line structures and the discontinuous NP chains parallel to the SHI beam. Numerical simulations of the optical absorption spectra suggested the NP chains but not the continuous line structures as the origin of the anisotropy. The optical anisotropy in CaF2 irradiated with SHIs is different from the shape elongation of NPs.

Magnetocrystalline anisotropy energy of AYbO3 (A= Sr, Ba and Ca) perovskite oxides

First-principles calculations and Monte Carlo simulations were employed to explore the effect of magnetic anisotropy on magnetocaloric properties and the magnetocrystalline anisotropy energy (MAE) of AYbO3 (A = Sr, Ba and Ca) perovskite oxides. The equilibrium lattice constant a0 is calculated AYbO3 (A = Sr, Ba and Ca). The elastic constants and related mechanical quantities such as bulk modulus B, Zener anisotropy factor A and Cauchy pressure Cp were calculated. CaYbO3 and SrYbO3 exhibit the half-metallic characteristic. While BaYbO3 exhibits semiconductor character. Sr, Ba and Ca atoms located in the A site affected the values of magnetocrystalline anisotropy energy (MAE) so changing the sign of the magnetic anisotropy constant in SrYbO3. The magnetic anisotropy shows that the easy axis is along the axis [100] for SrYbO3, CaYbO3 and BaYbO3 respectively. The critical temperature around Tc = 65 K using the MCS method. The isothermal magnetic entropy change was calculated for BaYbO3 under different magnetic fields and magnetic anisotropy. The highest isothermal entropy change obtained at H = 5T and D = 0 is 5.51 J/kg.K. While it’s equal to 8.10 J/kg.T at H = 5T and D =5.

Pr-modified microstructure enhances coercivity in Sm2Co17-type magnets

Optimizing components of the cellular structure is a strategy to enhance the magnetic performance of Sm2Co17-type magnets. In this study, Pr is used to modify the cellular structure in Sm2Co17-type magnets, achieving synergistic optimization of coercivity from 32.12 kOe to 34.71 kOe and remanence from 11.23 kGs to 11.36 kGs. Analyses reveal that the increased remanence is attributed to the enhanced saturation magnetization and better grain alignment in the Pr-5 magnet. The incomplete cellular structures and Cu-lean region near grain boundaries are ameliorated in the Pr-5 magnets. Meanwhile, the Pr-modified and Cu-enriched cellular boundary phase, which has a lower magnetocrystalline anisotropy field, enhances the pinning strength of the domain-wall and contributes to the increase in coercivity. The substitution of Pr for Sm has a lower substitution energy in SmCo5 than in Sm2Co17, leading to the slight enrichment of Pr in the 1:5H phase. However, excessive Pr leads to increased amounts of oxides and deteriorates magnetic properties. This work provides valuable insights into developing high-performance Sm2Co17-type magnets by optimizing the microstructure and microchemical components.

Intrinsic Magnetocrystalline Anisotropy Induced 3m-Symmetry Dependent Field-Free Switching in Epitaxial Garnet Films.

The electrical control of perpendicular magnetization without the need for external magnetic fields holds significant potential for next-generation spintronic devices. In this Letter, we have identified a 3m-symmetry dependent field-free switching phenomenon in (111)-oriented Tm_{3}Fe_{5}O_{12} single-crystal films capped with a platinum (Pt) layer. We demonstrate that this distinctive property arises due to the spontaneous breaking of mirror symmetry in magnetocrystalline anisotropy (MCA) for (111)-oriented magnetic films with a cubic structure, which results in a local out-of-plane MCA effective field with a 3m-symmetry dependence on the azimuth angle when the magnetization lies in the (111) plane. The observed field-free switching, facilitated by this MCA effective field, is validated by numerical simulations based both on macro-spin model and on micromagnetic theory. Our findings not only underscore the instrumental role of intrinsic MCA in enabling field-free magnetization switching but also enrich the fundamental understanding of the underlying switching dynamics.

Giant Uniaxial Magnetocrystalline Anisotropy in SmCrGe3.

Magnetic anisotropy is a crucial characteristic for enhancing the spintronic device performance. The synthesis of SmCrGe3 single crystals through a high-temperature solution method has led to the determination of uniaxial magnetocrystalline anisotropy. Phase verification was achieved by using scanning transmission electron microscopy (STEM), powder, and single-crystal X-ray diffraction techniques. Electrical transport and specific heat measurements indicate a Curie temperature (TC) of approximately 160 K, while magnetization measurements were utilized to determine the anisotropy fields and constants. Curie-Weiss fitting applied to magnetization data suggests the contribution of both Sm and Cr in the paramagnetic phase. Additionally, density functional theory (DFT) calculations explored the electronic structures and magnetic properties of SmCrGe3, revealing a significant easy-axis single-ion Sm magnetocrystalline anisotropy of 16 meV/fu. Based on the magnetization measurements, easy-axis magnetocrystalline anisotropy at 20 K is 13 meV/fu.

Importance of magnetic shape anisotropy in determining triaxial magnetic anisotropy of ferromagnetic semiconductor CrSCl monolayer

Abstract Magnetic anisotropy (MA) is pivotal for stabilizing long-range magnetic order in two-dimensional (2D) systems against thermal fluctuations. Here, we conduct a comprehensive investigation of the electronic and magnetic properties of CrSCl monolayer using first-principles methods and Monte Carlo (MC) simulations. Our results reveal that CrSCl monolayer exhibit a direct band gap ferromagnetic semiconductor (FMS) with a high Curie temperature (TC, 143 K). Notably, we identify triaxial magnetic anisotropy in this monolayer, characterized by the easy magnetization axis along the y-axis, intermediate axis along the x-axis, and hard axis along the z-axis. This anisotropy arises from a combination of magnetocrystalline anisotropy and shape anisotropy, in which shape anisotropy dominating over weak magnetocrystalline anisotropy. Orbital projection analysis shows that the major contribution of magnetic anisotropy energy comes from the d orbital of Cr atom. These findings provide some insights into the strain response of MA and suggest that studies of other FM monolayers may uncover future contenders for strain-switchable and ultra-compact spintronics devices.

Higher-order exceptional points and enhanced polarization sensitivity in anisotropic photonic time-Floquet crystals

We realized fourth-order exceptional points (EP-4s) in the quasienergy bands of anisotropic photonic time-Floquet crystals (APTCs), where the full in-plane permittivity tensors are periodically modulated in time. We developed the Floquet matrix method for APTCs, which provides a comprehensive study of the generation and characteristics of EP-4s. It is shown that an EP-4 is formed by the coalescence of three second-order exceptional points (EP-2s), which are classified into two types based on the band slopes near them. Additionally, the helicities of the four coalescing bands converge rapidly to zero at any given time when approaching the EP-4, following the ϵ1/4-dependence. Therefore, a strong polarization sensitivity is observed at the EP-4, which could inspire applications relevant to electromagnetic spins in APTCs.

Study of the influence of nitrogen doping on magnetic anisotropy in CoFe/MgO thin films with different deposition sequences

Abstract The ferromagnetic (FM)/MgO structure multilayers with perpendicular magnetic anisotropy (PMA) play a crucial role in magnetic random-access memory. It is essential to explore the methods for modulating magnetic anisotropy and to clarify the underlying mechanisms. We study the regulation of the magnetic anisotropy with nitrogen (N) doping concentration (CN) in two samples: Ta/CoFe(N)/MgO/Ta (S1) and Ta/MgO/CoFe(N)/Ta (S2) with different deposition sequences. The S1 sample exhibits in-plane magnetic anisotropy (IMA) without N doping. And the sample presents PMA when CN = 15%. The S2 sample shows weak PMA without N doping, and the sample converts to IMA when CN = 15%. X-ray photoelectron spectroscopy is performed to evaluate the oxidation level of Fe at the CoFe/MgO interface by the peak area ratio of Fe oxide to Fe (ϵ). It is believed that excessively high and low ϵ values correspond to over-oxidation and under-oxidation of Fe, respectively. For films deposited in different sequences, both over-oxidation and under-oxidation lead to IMA, while moderate oxidation facilitates the formation of PMA. The microstructure analysis reveals that the N doping does not affect the crystallization of the films, indicating that magnetocrystalline anisotropy has a minor influence on the changes of K eff.

Origin of magnetism via cation vacancy defects in non-stoichiometric NiCo2O4 nanoparticles: A synchrotron-based x-ray absorption and x-ray magnetic circular dichroism spectroscopic study

The present study probes the effect of cation distributions in the structural, optical, electronic, and magnetic properties of mixed-valent inverse-spinel NiCo2O4 (NCO) nanoparticles (NPs). NCO NPs were prepared using the sol–gel combustion method and the grain size was obtained in the magnetic exchange length range assumed to be from single-ion anisotropy. The Raman and photoluminescence spectroscopies confirm the presence of an inverse-spinel structure with different oxidation states, and vibrating sample magnetometry clarifies the existence of ferromagnetism with the presence of magnetic anisotropy among the cations. These NPs annealed at a higher grain-growth temperature accumulate ferrimagnetic properties and produce magneto-crystalline anisotropy making NCO an assuring material for spintronic applications. A detailed x-ray absorption spectroscopy and x-ray magnetic circular dichroism studies reveal an indestructible correlation between the distribution of the present cations and the element-specific origin of ferrimagnetic behavior. Ni is found to be accountable for the magnetic moment and electronic conduction, whereas Co is associated mainly with the generation of the magnetic anisotropy even in the polycrystalline NP form. This describes the anti-ferromagnetic coupling between Co and Ni ions that is pivotal in demonstrating the exchange interaction between these cations. The above result signifies the site-dependent cation valence states for the magnetic properties, and the extent of growing conditions are related to such cation-site dysfunction. This depicts further tunability in NCO as a functional oxide material.

Tunning Pr to achieve ultra-high magnetic properties of anisotropic nanocrystalline (Sm,Pr)Co5 magnets

In this study, we reported on the bulk anisotropic nanocrystalline Sm1-xPrxCo5 (x = 0.2–0.6) magnets aimed at achieving excellent energy products. Through a combination of advanced processing techniques, including melt spinning, hot pressing, and hot deformation process, bulk magnets with a fine nanocrystalline structure were successfully synthesized. The results revealed that the substitution of Pr strengthened the c-axis texture and the saturation magnetization, facilitating the enhancement of remanence (Mr) and maximum magnetic energy product [(BH)max]. However, Pr substitution decreased the coercivity (Hci) due to the reduced magnetocrystalline anisotropy field. Excellent magnetic properties are obtained for Sm0.4Pr0.6Co5 magnets with the (BH)max of 23.2 MGOe and Hci of 11.8 kOe. The prepared (Sm,Pr)Co5 magnets with different Pr contents have essentially the same microstructure with RECo5 main phase grains of about 400 nm in diameter and fine dispersed rare earth-rich nanoprecipitates. Our findings can provide reliable reference for manufacturing nanocrystalline permanent magnet materials and promote the development of rare earth permanent magnet new materials.

Statistical properties of circular and rectangular multi-sinc Schell-model beams propagating in uniaxial crystals

With the extended Huygens–Fresnel principle, we derive the expressions for the spectral intensity, coherence, and effective beam width of circular and rectangular multi-sinc Schell-model (MSSM) beams propagating through uniaxial crystals. Numerical simulations are employed to extensively explore how beam and crystal parameters modulate the optical field. The results reveal that the propagating field exhibits multiple ring-shaped and array-like intensity distributions, with adjustable features such as the number of concentric rings, central brightness, array dimensions, and the morphology and diversity of sub-beams. Additionally, the spectral coherence displays an oscillatory distribution that evolves into a Gaussian distribution as the transmission distance increases. The anisotropy of uniaxial crystals not only influences the morphology of intensity distribution but also affects the evolution rate of coherence and the expansion rate of effective beam width. Our work contributes to optimizing beam propagation through uniaxial crystals, potentially benefiting precision optical systems in laser technology.

Temperature effect on acoustic waves reflection condition in anisotropic crystals

The reflection of acoustic waves in anisotropic media has found important applications in acousto-optics (AO). It may be realized both with changing the type of acoustic mode and without it. The latter case is applied for the realization of the quasi-collinear AO diffraction geometry for which the incident light and ultrasound wave group velocity vectors are collinear. The acoustic wave involved into the AO interaction exists due to the reflection from the AO cell input optical face. Quasi-collinear configuration of AO interaction enables achieving exceptionally high spectral resolution and diffraction efficiency, therefore, it is applied for the laser pulse shaping. However, the application of AO tunable filters for such purposes requires high acoustic power values, which induce the temperature gradients in the AO crystal. Temperature variations influence the stiffness constants of the AO crystal, impacting the characteristics of incident and reflected acoustic waves, affecting the acoustic wave reflection condition and consequently, the AO diffraction characteristics. This study presents the theoretical and experimental examination of the temperature influence on the acoustic beam reflection in anisotropic crystals on the example of tellurium dioxide crystal. This paper is supported by the Russian Science Foundation, grant 23-12-00057.

Internal radiation effect on semiconductor β-Ga2O3 crystals grown by the VB Method and anisotropic thermal stress

Gallium oxide crystals are semitransparent semiconductors with good optical and electrical properties, which allow their use for several technological applications. During the growth process of β-Ga2O3 crystals, internal radiation plays a crucial role that affects the growth process and then the crystal quality. In this work, the effect of the melt and the crystal transparency on the vertical Bridgman growth of β-Ga2O3 oxide is thoroughly studied. Using a global 2D/3D finite element model, temperature, melt flow, melt-crystal interface, and three-dimensional anisotropic thermal stress are computed at different growth stages. At each stage, four cases are considered, namely, opaque melt and crystal, semi-transparent melt and opaque crystal, semitransparent crystal and opaque melt, and finally semitransparent melt and crystal. The role of internal radiation in each case at different growth stages is then highlighted separately and then coupled together. It was found that the melt-crystal interface is shifted from a convex shape at the early stage to a nearly plane and then to a concave shape at the last stage. The melt flow is then changed from two rolls pattern at the beginning to a single-roll structure at the last stage. Thermal stress of the as-grown ingot is decreased during the growth due to the decrease of temperature non-linearities. Internal radiation inside the crystal acts to increase the melt-crystal interface convexity at the early and middle stages of the growth process and leads to a decrease in its concavity at the final stage. However, the melt transparency leads to the opposite effects, i.e., it decreases the interface convexity at the early stage and increases the interface concavity at the final stage. As a result, for semitransparent crystal and melt, the interface is between the two previous cases. The calculated thermal stresses are found to be more affected by the transparency of the crystal than the melt as the absorption coefficient of β-Ga2O3 crystal is smaller than that of the melt. At all stages, the thermal stresses are found to be larger for the opaque case due to the increase of temperature non-linearities in the crystal. Large values are found at the bottom and the lower part of the periphery. Furthermore, internal radiation inside the melt plays a major role during the early growth stage due to the large liquid volume. It reduces the melt flow intensity close to the free surface where the shear stress is combined with the buoyant force and leads to the flattening of the interface decreasing then the radial temperature gradients, which leads to small attenuation of the thermal stress. At the final stage, the transparency of the melt plays the opposite role due to the increase of the interface concavity which leads to an increase in both of the temperature gradients and the thermal stress inside the crystal. Due to the crystal anisotropy, and especially to the large thermal expansion coefficient in the [010] direction, the 3D thermal stress values are found to be larger in this direction, especially in the bottom part of the ingot. Comparisons with available experimental and numerical works are provided in this paper.

The magnetocrystalline anisotropy of Y2(FexCo1-x)17Hy compounds

The article presents the results of magnetocrystalline anisotropy (MCA) investigation of Y2(FexCo1-x)17 (x=0.03-0.50) rare-earth quasi-binary bulk crystal samples and their hydrides. The specific magnetization curves m(H) were measured in the wide temperature range for the original Y2(FexCo1-x)17 samples and for the corresponding hydrides. Magnetocrystalline anisotropy constants K1, K2 were calculated, temperature and compositional dependence curves were plotted based on m(H) data. The temperature dependences of specific saturation magnetization ms(T) and first anisotropy constant K1(T) were analyzed to determine temperature behavior patterns, estimate the Curie temperature Tc and predict the spin-reorientation transitions. The first anisotropy constant, saturation magnetization and Curie temperature compositional dependences of original Y2(FexCo1-x)17 and hydrogenated Y2(FexCo1-x)17Hy samples were compared to identify the hydrogenation effect on the magnetic properties of investigated compounds. The revealed patterns were explained in terms of the crystal structure of the compounds.

From [Ba3S][GeS₄] to [Ba₃CO₃][MS₄] (M = Ge, Sn): Enhancing optical anisotropy in IR birefringent crystals via functional group implantation

Birefringent crystals are crucial for manipulating light's phase and polarization, making them vital components in various optical devices. Traditionally, strategies for designing high-performance birefringent crystals have focused on modifying the parent structure. However, there are limited examples demonstrating how changing functional groups can effectively enhance birefringence (Δn), as such changes often significantly alter the crystal structure. In this study, we propose a “functional group implantation” strategy aimed at significantly improving birefringent performance within the chalcogenide system. This involves replacing the isotropic [S]2⁻ ions with anisotropic π-conjugated [CO₃]2⁻ groups. We validated this approach through comprehensive comparisons between the chalcogenide [Ba₃S][GeS₄] and the oxychalcogenide [Ba₃CO₃][MS₄] (where M ​= ​Ge and Sn), both of which adopt the same space group and feature the same arrangements of functional groups. Experimental characterization and theoretical calculations confirm that the [CO₃]2⁻ groups exhibit significantly greater polarization anisotropy than the [S]2⁻ groups. This difference leads to a marked increase in Δn in [Ba₃CO₃][MS₄] (ranging from 0.088 to 0.112 ​at 546 ​nm) compared to [Ba₃S][GeS₄] (0.021 ​at 546 ​nm). This finding not only broadens the structural chemistry of π-conjugated chalcogenides but also illustrates the potential of functional group implantation for designing infrared birefringent crystals with enhanced optical anisotropy.

A first-principle study on the two dimensional Janus MXene TaFeC with spin gapless behaviour

Based on density functional theory calculations, we have investigated the two-dimensional Janus MXene TaFeC, focusing particularly on the magnetic and electronic properties. It is found TaFeC is dynamically, mechanically, and thermodynamically stable. With both sizable Curie temperature and enhanced out-off plane magneto-crystalline anisotropy energy, the Janus MXene TaFeC has the potential to be applied at finite temperature. The mechanism behind the magneto-crystalline anisotropy energy and magnetism are explained based on perturbation theory and crystal field splitting, respectively. Under biaxial strain, the ferromagnetic order is robust stable. With a various of effective potential on Fe, the electronic structure evolution is studied, where the effective potential of 1.3 eV is a special case. In such case, the Janus MXene TaFeC can be classified as a new type of spin-gapless semiconductor beyond the previous classifications, behaving topological nontrivial edge states.

Electromagnetic wave absorption properties of La-Ca-Co-Al substituted M-type Sr-hexaferrite-rubber sheets in 50–75 GHz range

This study reports on the electromagnetic (EM) wave absorption characteristics of M-type hexaferrite-rubber sheets, where Al was additionally substituted into high-performance permanent magnet compositions, Sr0.2La0.4Ca0.4Co0.25Fe9.75-xAlxO19-d (LCCA-SrM, x = 0, 0.2, 0.4). While non-substituted SrFe12O19 powder-rubber sheet had a ferromagnetic resonance frequency (fFMR) of 48.5 GHz, the LCCA-SrM (x = 0, 0.2, 0.4) rubber sheets showed increased fFMR values of 63.3, 66.2, and 68.5 GHz, respectively, with the increasing x values. The frequency of minimum reflection loss (fRLmin) also increased correspondingly with the change in fFMR. This significant increase in fFMR and fRLmin is attributed to the greatly enhanced magnetocrystalline anisotropy from the La-Ca-Co substitution in SrM and the incremental increase in magnetic anisotropy field (Hani) due to Al substitution. The absorption band (Δf) satisfying RL < −10 dB, was 45.0–53.4 GHz for SrM, and 52.9–75, 50–75, and 59–74.2 GHz for the x = 0, 0.2, and 0.4 samples, respectively. The x = 0.2 sheets demonstrated the widest absorption, covering 50–75 GHz. This study is highly promising as it is the first to demonstrate the feasibility of developing an EM wave absorber sheet for frequencies above 50 GHz using hexaferrite powder synthesized via the solid-state method, which offers scalability and cost-efficiency for mass production.

Composition and microstructure engineering of Fe–Si–Co soft magnetic alloys with enhanced performance

The growing demand for high-efficiency and low-loss energy conversion and transportation techniques urges the development of advanced Fe–Si based soft magnet alloys. Simultaneous achievement of low coercivity (Hc) and large saturation magnetization (Ms) however, remains challenging. In this study, soft magnetic alloys with the composition Fe82–xSi18Cox (x = 0 at.%, 4 at.%, 8 at.%, 12 at.%, 16 at.%, and 20 at.%) have been designed followed by microstructural tuning. The Co incorporation results in initially decreased Hc followed by increment due to reduced magnetocrystalline anisotropy and increased saturation magnetostriction from negative to positive values of the alloys. Meanwhile, the Ms raises with subsequent reduction, which origins from competitive mechanisms of increased average moment of Fe atoms and decreased average moment of Co atoms according to first principles calculations. Microstructural evolution during annealing of the Fe70Si18Co12 with synergistically optimized Hc and Ms has been revealed that after elevated-temperature annealing, the DO3 phase is predominately transformed from the B2 phase accompanied by an increase in the degree of ordering. The growth of the DO3 phase deteriorates the Hc due to the aggravating pinning effect on the domain wall movement, which arises from the inhomogeneous magnetization distribution caused by increasing antiphase boundaries.