Wall Pinning Research Articles

Numerical Simulation of Electromagnetic Nondestructive Testing Technology for Elasto–Plastic Deformation of Ferromagnetic Materials Based on Magneto–Mechanical Coupling Effect

A numerical tool for simulating the detection signals of electromagnetic nondestructive testing technology (ENDT) is of great significance for studying detection mechanisms and improving detection efficiency. However, the quantitative analysis methods for ENDT have not yet been sufficiently studied due to the absence of an effective constitutive model. This paper proposed a new magneto–mechanical model that can reflect the dependence of relative permeability on elasto–plastic deformation and proposed a finite element–infinite element coupling method that can replace the traditional finite element truncation boundary. The validity of the finite element–infinite element coupling method is verified by the experimental result of testing electromagnetic analysis methods using TEAM Problem 7. Then, the reliability and accuracy of the proposed model are verified by comparing the simulation results under elasto–plastic deformation with experimental results. This paper also investigates the effect of elasto–plastic deformation on the transient magnetic flux signal, a quantitative hyperbolic tangent model between Bzpp (peak–peak value of the normal component of magnetic flux signal) and elastic stress, and the exponential function relationship between Bzpp and plastic deformation is established. In addition, the difference and mechanism of a magnetic flux signal under elasto–plastic deformations are analyzed. The results reveal that the variation of the transient magnetic flux signal is mainly due to domain wall pinning, which is significantly affected by elasto–plastic deformation. The results of this paper are important for improving the accuracy of quantitative ENDT for elasto–plastic deformation.

ProtonationDriven Polarization Retention Failure in Nano-Columnar Lead-Free Ferroelectric Thin Films.

Understanding microscopic mechanisms of polarization retention characteristics in ferroelectric thin films is of great significance for exploring unusual physical phenomena inaccessible in the bulk counterparts and for realizing thin-film-based functional electronic devices. Perovskite (K,Na)NbO3 is an excellent class of lead-free ferroelectric oxides attracting tremendous interest thanks to its potential applications to nonvolatile memory and eco-friendly energy harvester/storage. Nonetheless, in-depth investigation of ferroelectric properties of (K,Na)NbO3 films and the following developments of nano-devices are limited due to challenging thin-film fabrication associated with nonstoichiometry by volatile K and Na atoms. Herein, ferroelectric (K,Na)NbO3 films of which the atomic-level geometrical structures strongly depend on thickness-dependent strain relaxation are epitaxially grown. Nanopillar crystal structures are identified in fully relaxed (K,Na)NbO3 films to the bulk states representing a continuous reduction of switchable polarization under air environments, that is, polarization retention failures. Protonation by water dissociation is responsible for the humidity-induced retention loss in nano-columnar (K,Na)NbO3 films. The protonation-driven polarization retention failure originates from domain wall pinning by the accumulation of mobile hydrogen ions at charged domain walls for effective screening of polarization-bound charges. Conceptually, the results will be utilized for rational design to advanced energy materials such as photo-catalysts enabling ferroelectric tuning of water splitting.

Time-dependent relaxation dynamics of remanent strain in textured PMN–PZ–PT and PMN–PIN–PT piezoceramics

Crystallographically textured lead-based piezoceramics, particularly Pb(Mg1/3Nb2/3)O3–PbZrO3–PbTiO3 (PMN–PZ–PT) and Pb(Mg1/3Nb2/3)O3–Pb(In1/2Nb1/2)O3–PbTiO3 (PMN–PIN–PT), play a pivotal role in electromechanical applications due to their enhanced field-induced strain responses. However, challenges in precision positioning arise from heightened remanent strain (Sr) caused by embedded templates in the textured grains, impacting strain cycle consistency due to increased time delay for Sr relaxation. This study investigates the time-dependent relaxation dynamics of Sr in textured PMN–PZ–PT and PMN–PIN–PT piezoceramics, focusing on experimentally determining relaxation time between cycles required for complete Sr relaxation. Our findings reveal that the relaxation time in textured piezoceramics is notably longer than that in their non-textured counterparts, primarily due to pinning of domain walls by embedded templates. This work provides insights into the strain relaxation characteristics of piezoceramics and underscores the importance of time delay in designing precision positioning systems for improved reliability.

Unusual Magnetocaloric Effect Triggered by Spin Reorientation

AbstractAdvancements and utilization of magnetic refrigeration technology hinge on the ongoing enhancement and optimization of magnetic refrigeration material properties. Nevertheless, the intricacy of the magnetocaloric effect (MCE) mechanism has emerged as a bottleneck, constraining the progress and refinement of magnetic refrigeration materials. In this study, a classic magnetic system is chosen to investigate the mechanism of MCE across four different scales–macroscopic magnetism, micrometer‐scale magnetic domains, atomic magnetic moments, and electronic structure. It simultaneously exhibits two inverse MCEs and one direct MCE, with a working temperature span as wide as 125 K (most are <50 K) for the direct MCE. The measurements of the vibrating sample magnetometer, in situ Lorentz electron microscopy and variable‐temperature neutron powder diffraction directly reveal that the complex magnetic entropy changes arise from the magnetic domain wall pinning, the instability of Ho magnetic moments, and the spin rotation. First‐principles calculations elucidate the crucial role of strong hybridization between localized Ho and itinerant Co electrons in the spin reorientation of HoCo4Al. This study contributes significantly to comprehending the induction mechanism of the MCE and holds vital reference value for exploring new magnetic refrigeration materials and enhancing MCE performance.

Anhysteretic strains in ferroelectric ceramics under electromechanical loading

This study investigates anhysteretic strains in PZT ceramics. The anhysteretic curves are associated with a stable balanced state of polarization in the domain structure, excluding dissipative effects related to mechanisms such as domain wall pinning. Anhysteretic measurements are representative of an -ideal- scenario in which the material would undergo no energy loss due to dissipative processes, focusing on the stable and reversible aspects of the domain configuration. The different methodologies employed to measure deformations under electromechanical loading are presented, leading to the introduction of digital image correlation (DIC) as the chosen technique, recognized for its ability to capture detailed information on transverse and longitudinal strain. The article then describes a procedure developed to obtain anhysteretic strain and anhysteretic polarisation for different levels of compressive loadings. The subsequent presentation of the results of the transverse and longitudinal strain analyses provides valuable insights into the reversible and irreversible behavior of the material. They can be used as a basis for the thermodynamical modelling approaches grounded on separating reversible and irreversible contributions or as a validation of existing models describing anhysteretic behavior. The compressive stress affects both the shape of hysteretic and anhysteretic curves. The anhysteretic curve represents a stable equilibrium in the domain structure. Compressive stress reduces strain by affecting the pinning of domain walls. These points justify the interest in studying the effect of compressive stress on the anhysteretic behavior of ferroelectrics.

Ferroelectric Al0.85Sc0.15N and Hf0.5Zr0.5O2 Domain Switching Dynamics.

The capability to reliably program partial polarization states with nanosecond programming speed and femtojoule energies per bit in ferroelectrics makes them an ideal candidate to realize multibit memory elements for high-density crossbar arrays, which could enable neural network models with a large number of parameters at the edge. However, a thorough understanding of the domain switching dynamics involved in the polarization reversal is required to achieve full control of the multibit capability. Transient current integration measurements are adopted to investigate the domain dynamics in aluminum scandium nitride (Al0.85Sc0.15N) and hafnium zirconium oxide (Hf0.5Zr0.5O2). The switching dynamics are correlated to the crystal structure of the films. The contributions of domain nucleation and domain wall motion are decoupled by analyzing the rate of change of the time-dependent normalized switched polarization. Thermally activated creep domain wall motion characterizes the Al0.85Sc0.15N switching dynamics. The statistics of independently nucleating domains and the domain wall creep motion in Hf0.5Zr0.5O2 are associated with the spatially inhomogeneous distribution of local switching field due to polymorphism, absence of preferential crystallite orientation, as well as defects and charges that can be located at the grain boundaries. The c-axis texture, single-phase nature, and strong likelihood of less fabrication process-induced defects contribute to the homogeneity of the local switching field in Al0.85Sc0.15N. Nonetheless, defects generated and redistributed upon bipolar electric field switching cycling result in Al0.85Sc0.15N domain wall pinning. The wake-up effect in Hf0.5Zr0.5O2 is explained thorough the continuous addition of switchable regions associated with two independent distributions of characteristic switching times.

Orientation dependent grain boundary precipitate distribution and the weakened pinning effects on domain walls in sintered Sm2Co17-type magnet

In a typical sintered multi-component Sm2Co17-type magnet with strong fiber texture, we show that the distributions of grain boundary precipitates (GBPs) are heavily dependent on the grain boundary (GB) geometry with respect to the texture direction, which has significant effects on domain wall pinning. Results demonstrate that the continuous GBPs turn into discrete upon the angle between {0001} planes and GB increases from 0° to 90°, meanwhile the GBPs thickness and precipitation free zones (PFZs) width both increase linearly by a factor of 2.5–4. Transmission electron microscopy (TEM) reveals that the GBPs are alternatively stacked Cu-rich SmCo5 and Zr-rich Smn+1Co5n−1 (n = 2,3,4) compounds, while the PFZs are composed of 2:17R and intermediate 2:17R’ phases. Atomic-level elemental mappings and first-principles calculations indicate that Cu exists at the Co-2c site in the SmCo5 forming SmCo3Cu2 and Zr locates at the dumbbell Sm-6c sites in the Smn+1Co5n−1. The symbiotic GBPs have orientation relationships of [0001]GBPs//[0001]2:17R and [101¯0]GBPs//[21¯1¯0]2:17R. The formation of anisotropic GBPs is owing to the strong fiber texture, i.e., the larger angle between {0001} planes and GBs, the more {0001} diffusion channels for atoms to GBs, resulting in discrete and thick GBPs. In-situ Lorentz TEM shows that the domain walls interrupted by GBPs migrate easily under applied magnetic fields. Possible approaches to enhance the magnetic hardness via tuning the GBs are proposed.

Nanocrystalline SmCo12 main-phase alloys with V-doping: Structure stability and magnetic performance

The intermetallic compounds based on the tetragonal ThMn12 prototype crystal structure have exhibited great potential as advanced rare-earth-lean permanent magnets due to their excellent intrinsic magnetic properties. However, the trade-off between the phase stability and the magnetic performance is often encountered in the ThMn12-type magnets. This work was focused on the effects of V doping and nanostructuring on the phase stability and magnetic properties of ThMn12-type Sm-Co-based magnets. Novel SmCo12-based nanocrystalline alloys with the SmCo12 main phase were prepared for the first time. The prepared alloys from the optimal design achieved obviously higher coercivity than the isotropic SmFe12-based alloys, together with comparable performance of other magnetic features. The enhancement in the coercivity was ascribed to the pinning of domain walls by the nanocrystalline grain boundaries and stacking faults. First-principles calculations and magnetic structure analysis disclosed that V substitution can stabilize the SmCo12 lattice and elevate its magnetocrystalline anisotropy. This study provides a new approach to developing stabilized metastable structured rare-earth-lean alloys with high magnetic performance.

Magnetic domain wall pinning and depinning in DyFeO3 single crystal

Several papers reported the experimental phenomena of single crystal orthoferrite DyFeO3 samples: the sample magnetization increases with increasing temperature, under a constant low magnetic field along the easy magnetizing direction of the samples, and then reaches a maximum value. These phenomena are better explained using magnetic domain wall pinning at low temperatures and depinning at high temperatures, which were neglected in previous papers. In this article, we argue that the magnetic domain wall motion must be taken into consideration in order to explain these experimental phenomena. Based on an O 2p itinerant electron model, we discussed the role of the magnetic domain and the characteristics of cation magnetic moments in relevant single crystal materials.

Polar orientation and extension in a novel crystallographic model for PbTiO3-based perovskites explaining the experimental ferroelectric thermal anomalies

PbTiO3-based ferroelectric solid-solution ceramics have been widely used for electromechanical devices. However, it is still challenging to separate and control the contributions to the electromechanical functionalities, mainly as a function of temperature, where thermal anomalies and phase transitions can be observed. This study investigates the ultrasonic velocity and attenuation and the dielectric, ferroelectric and structural features of Pb0.55Ca0.45TiO3 ceramics from low temperatures (10 or 115 K) up to room temperature as an example of A-site isovalent substitution in PbTiO3. Such a combination of information makes possible the phenomenological deconvolution of the effects of ferroelectric domain wall pinning and structural features on spontaneous electric polarization. The room-temperature symmetry was determined as Pna21. The results show that this model refined by the Rietveld method for synchrotron X-ray diffraction patterns from 115 K to room temperature can explain the polarization extension features of these materials during heating. This study shows a correlation between structural thermal anomalies and low-temperature electric polarization in PbTiO3-based ferroelectric ceramics.

Domain wall magnetic tunnel junction-based artificial synapses and neurons for all-spin neuromorphic hardware

We report a breakthrough in the hardware implementation of energy-efficient all-spin synapse and neuron devices for highly scalable integrated neuromorphic circuits. Our work demonstrates the successful execution of all-spin synapse and activation function generator using domain wall-magnetic tunnel junctions. By harnessing the synergistic effects of spin-orbit torque and interfacial Dzyaloshinskii-Moriya interaction in selectively etched spin-orbit coupling layers, we achieve a programmable multi-state synaptic device with high reliability. Our first-principles calculations confirm that the reduced atomic distance between 5d and 3d atoms enhances Dzyaloshinskii-Moriya interaction, leading to stable domain wall pinning. Our experimental results, supported by visualizing energy landscapes and theoretical simulations, validate the proposed mechanism. Furthermore, we demonstrate a spin-neuron with a sigmoidal activation function, enabling high operation frequency up to 20 MHz and low energy consumption of 508 fJ/operation. A neuron circuit design with a compact sigmoidal cell area and low power consumption is also presented, along with corroborated experimental implementation. Our findings highlight the great potential of domain wall-magnetic tunnel junctions in the development of all-spin neuromorphic computing hardware, offering exciting possibilities for energy-efficient and scalable neural network architectures.

Grain boundary diffusion behaviors of hot-deformed magnets in different directions with additional pressure

The diffusion behaviors of hot-deformed magnets were investigated in two different directions with additional pressure. For magnets diffusion along the direction parallel to the C-axis (∥C magnets) and magnets diffused perpendicular to the C-axis (⊥C magnets), the coercivity increased by 3.66 kOe and 4.94 kOe, while the remanence decreased by 1.39 kGs and 1.04 kGs, respectively. The coercivity and remanence are both higher in ⊥C magnets as compared to ∥C magnets. SEM indicates that the orientation in ⊥C magnets is better than in ∥C magnets, which contributes to the more excellent remanence. The SEM also suggests the distribution of the rare-earth-rich phase was more uniform in ⊥C magnets. EPMA shows that the diffusion depth of ⊥C magnets reaches around 1500 μm, while ∥C magnets is less than 1250 μm. Both the homogeneous distribution of the rare-earth-rich phase and deeper diffusion depth are responsible for the improved coercivity of ⊥C magnets. EDS shows that the Nd70Cu30 preferentially entered the interior of the magnets through the interface formed by the fine grains of melt-spun ribbon edges for ∥C magnet during grain boundary diffusion process. But the ⊥C magnet did not show this preference during diffusion. The grain morphology and diffusion resistance of magnets in different directions is the fundamental cause for the difference of Cu distribution in ∥C magnet and ⊥C magnet. The initial magnetization curves suggests that the coercivity mechanism of the initial magnets combines nucleation and domain wall pinning, while the domain wall pinning is enhanced for the diffused magnets. The optimum performances with Br = 13.21 kGs, Hcj = 18.23 kOe, and (BH)max = 43.01 MGOe were achieved in ⊥C magnets.

Effect of heavy Dy3+ doping on the magnetic, structural, morphological, and optical characteristics of CuDyxFe2-xO4 nanoparticles

The present study explored the impacts of dysprosium (Dy³⁺) doping on the structure, optical, and magnetic characteristics of CuDyxFe2-xO4 nanospinel ferrites synthesized via a hydrothermal method. X-ray diffraction revealed a tetragonal spinel matrix with increasing Dy³⁺ content, accompanied by a secondary Dy2O3 phase at higher dopant levels for x ≥ 0.1. Crystallite size and microstrain increased with doping, while infrared spectroscopy confirmed the spinel structure and distinct vibrational modes. Interestingly, saturation magnetization displayed a non-monotonic trend, first rising due to spin polarization and cation redistribution, then decreasing with higher Dy³⁺ content attributable to the Dy2O3 phase and larger grains. Coercivity exhibited a maximum at x = 0.2, reflecting the interplay between magnetocrystalline anisotropy, canting angles, and domain wall pinning. These findings highlight the complex interplay between Dy³⁺ doping and the resulting properties, paving the way for tailored CuDyxFe2-xO4 nanospinel ferrites functionalities. . Further research involving advanced techniques and targeted dopant levels holds promise for optimizing properties for diverse technological applications such as magnetic storage, microwave devices, sensors, and hyperthermia treatment.

Effects of Mn doping on the conduction mechanism and dielectric nonlinearity of Na0.5Bi0.5TiO3 thin film

In this work, the conduction mechanism and dielectric nonlinearity of undoped and Mn-doped Na0.5Bi0.5TiO3 (NBT) films were investigated. The potential conduction mechanism in relatively low electric fields should be dominated by hopping conduction rather than typical Ohmic conduction. In the high electric field region, the conduction mechanism is dominated by Poole–Frenkel emission and Schottky emission. The enhancement of electrical insulation of NBT films after Mn doping was shown to result from a decrease in oxygen vacancies and elevation of the conduction energy barrier. Furthermore, significant improvements in the nonlinearity of both the dielectric constant and polarization of Mn-doped NBT films were observed, as indicated by the results of Rayleigh fitting and first-order reversal curve distributions. Such enhancements were attributed to the reduction in domain wall pinning, decreased interference from electrostatic potential and improved leakage characteristics.

Understanding the Effect of Curvature on the Magnetization Reversal of Three-Dimensional Nanohelices.

Comprehending the interaction between geometry and magnetism in three-dimensional (3D) nanostructures is important to understand the fundamental physics of domain wall (DW) formation and pinning. Here, we use focused-electron-beam-induced deposition to fabricate magnetic nanohelices with increasing helical curvature with height. Using electron tomography and Lorentz transmission electron microscopy, we reconstruct the 3D structure and magnetization of the nanohelices. The surface curvature, helical curvature, and torsion of the nanohelices are then quantified from the tomographic reconstructions. Furthermore, by using the experimental 3D reconstructions as inputs for micromagnetic simulations, we can reveal the influence of surface and helical curvature on the magnetic reversal mechanism. Hence, we can directly correlate the magnetic behavior of a 3D nanohelix to its experimental structure. These results demonstrate how the control of geometry in nanohelices can be utilized in the stabilization of DWs and control of the response of the nanostructure to applied magnetic fields.

Magnetization reversal and domain wall pinning in nanowires with nonlinear modulation of diameter

Micromagnetic simulation of magnetization reversal is studied on cylindrical tapered nanowire with radii 50nm and 10nm at wider and narrower end respectively. The radius decreases from wider end to the narrower end non-linearly as R-βzδ where δ is the degree of non-linearity. Value of δ is either an integer (n) or reciprocal of integer (1/n). At remanence, a vortex and flower state at the wider and narrower extremes respectively are exhibited combined with ferromagnetic ordering in the majority of volume. When δ is below 1/2, pinning and depinning of domain wall drives the hysteresis following a Modified Kondorsky model. Otherwise, the magnetization reversal proceeds with nucleation and propagation of vortex domain wall according to Kondorsky model. The propagation of domain wall is dependent upon the value of δ which suggests tunability of magnetic characteristics with modulation of diameter. For instance, large values of coercivity is observed for wires that have strong domain wall pinning which is considerably reduced when depinning and propagation of domain wall takes over. A large range (20kA/m to 130kA/m) of coercivity can be obtained by changing the modulation.

Direct visualization of domain wall pinning in sub-100 nm 3D magnetic nanowires with cross-sectional curvature.

The study of 3D magnetic nanostructures has uncovered rich phenomena including the stabilization of topological spin textures using nanoscale curvature, controlled spin-wave emission, and novel ground states enabled by collective frustrated interactions. From a technological perspective, 3D nanostructures offer routes to ultrahigh density data storage, massive interconnectivity within neuromorphic devices, as well as mechanical induction of stem cell differentiation. However, the fabrication of 3D nanomagnetic systems with feature sizes down to 10 nm poses a significant challenge. Here we present a means of fabricating sub-100 nm 3D ferromagnetic nanowires, with both cross-sectional and longitudinal curvature, using two-photon lithography at a wavelength of 405 nm, combined with conventional deposition. Nanostructures with lateral features as low as 70 nm can be rapidly and reproducibly fabricated. A range of novel domain walls, with anti-vortex textures and hybrid vortex/anti-vortex textures are enabled by the cross-sectional curvature of the system, as demonstrated by micromagnetic simulations. Magnetic force microscopy experiments in an externally applied magnetic field are used to image the injection and pinning of domain walls in the 3D magnetic nanowire. At specific field values, domain walls are observed to hop from trap to trap, providing a direct means to probe the local energy landscape.

Investigation into the role of Co doping in the microstructure and magnetic characteristics of nanocrystalline Nd-Ce-La-Fe-B alloys

In this study, [Nd0.5(Ce0.5La0.5)0.5]17Fe78-xCoxB6 (x = 0 ∼ 1.5 at. %) ribbons are prepared via a direct melt-spun (wheel speed of 28 m/s). The impacts of Co doping on the crystal structure, Curie temperature Tc, magnetic parameters (Br, Hci and (BH)max) of [Nd0.5(Ce0.5La0.5)0.5]17Fe78-xCoxB6 alloys are systematically assessed. X-ray diffraction patterns and Rietveld refinement data demonstrate that the exclusive presence of a pure hard magnetic 2:14:1 phase without any secondary phases across all samples. Compared with Co-free sample (x = 0), the Br, Hci and (BH)max of Co-doped with x = 1 are increased by 15.0 %, 4.6 % and 42.9 %, respectively. The coercivity mechanism in the [Nd0.5(Ce0.5La0.5)0.5]17Fe78-xCoxB6 alloys arises from the integration of reversed domain nucleation and robust pinning of domain walls. The magnetic domain structure and quantitative analysis based on the microstructure and demagnetization behavior are conducted to investigate the exchange coupling interactions among the 2:14:1 grains. This exploration reveals that the enhanced magnetic properties stem from the refinement of the 2:14:1 phase's grains, leading to more uniform and smaller interaction domains.

Bias-polarity-dependent asymmetric domain switching in fatigued epitaxial BiFeO3 (001) capacitors

We investigated domain switching dynamics in pristine and fatigued epitaxial BiFeO3 (001) capacitors using piezoresponse force microscopy and switching current measurements. In contrast to domain growth-dominant switching in the pristine capacitors, we observed two different bias-polarity-dependent switching dynamics in the fatigued capacitors. Under a positive bias on the top electrode of the fatigued capacitors, the domain walls were strongly pinned, indicating that nucleation played a critical role in domain switching. However, when a negative bias was applied, the domain wall velocity decreased owing to weak pinning of the domain walls, but domain growth remained the governing process of the domain switching behavior. The intriguing asymmetric domain switching and related domain wall pinning behavior in the fatigued BiFeO3 capacitors may be attributed to the formation of a defect layer near the interface between the ferroelectric film and electrode.

Micromagnetics and multiscale hysteresis simulations of permanent magnets

PurposeConfronting the unveiled sophisticated structural and physical characteristics of permanent magnets, notably the samarium–cobalt (Sm-Co) alloy, This work aims to introduce a simulation scheme that can link physics-based micromagnetics on the nanostructures and magnetostatic homogenization on the mesoscale polycrystalline structures.Design/methodology/approachThe simulation scheme is arranged in a multiscale fashion. The magnetization behaviors on the nanostructures examined with various orientations are surrogated as the micromagnetic-informed hysterons. The hysteresis behavior of the mesoscale polycrystalline structures with micromagnetic-informed hysterons is then evaluated by computational magnetostatic homogenization.FindingsThe micromagnetic-informed hysterons can emulate the magnetization reversal of the parameterized Sm-Co nanostructures as the local hysteresis behavior on the mesostructures. The simulation results of the mesoscale polycrystal demonstrate that the demagnetization process starts from the grain with the largest orientation angle (a) and then propagates to the surrounding grains.Research limitations/implicationsThe presented scheme depicts the demand for integrating data-driven methods, as the parameters of the surrogate hysteron intrinsically depend on the nanostructure and its orientation. Further hysteron parameters that help the surrogate hysteron emulate the micromagnetic-simulated magnetization reversal should be examined.Originality/valueThis work provides a novel multiscale scheme for simulating the polycrystalline permanent magnets’ hysteresis while recapitulating the nanoscale mechanisms, such as the nucleation of domains, and domain wall migration and pinning. This scheme can be further extended to simulate the part-level hysteresis considering the mesoscale features.