Stress‐Mediated Ferromagnetic Spinodal Decomposition for Giant Low‐Field Magnetostriction

ABSTRACT

Nonequilibrium stress can trigger unconventional domain dynamics in magnetoelastic systems. Using phase‑field simulations, we reveal a stress‑driven spinodal decomposition in rhombohedral (R) Tb_0.3Dy_0.7Fe₂ (Tb–Dy–Fe) alloys with <111> easy axes. Gradual unloading of a [001] compressive stress induces a spontaneous, fully reversible transition from a uniform microdomain (MD) to an alternating nanodomain‑in‑microdomain (NIMD) pattern. Unlike classical nucleation‑and‑growth processes, this transformation proceeds via global, continuous small‑angle spin reorientation. Mechanistically, it originates from the negative curvature of the effective magnetic anisotropy (EMA) energy landscape, a direct analogue of thermodynamic spinodal instabilities. Magnetic exchange interactions oppose the decomposition, akin to chemical gradient energies, whereas magnetocrystalline anisotropy serves as the primary driving force. Remarkably, the resulting NIMD architecture delivers an ultrahigh magnetostriction of 721.5 ppm under a weak magnetic field of only 19.5 mT. This exceptional low‑field performance stems from the unique ability of R‑type nanodomains to undergo domain‑wall displacement, amplifying lattice deformation through energy minimization. Our findings establish a stress‑mediated magnetization control paradigm, opening new avenues for adaptive magnetoelastic materials in low‑field sensors and stress‑responsive nanotechnologies.

Stress‑driven ferromagnetic spinodal decomposition transforms uniform microdomains into alternating nanodomain patterns in the single rhombohedral phase of Tb–Dy–Fe. This spontaneous and reversible process originates from the negative curvature of the magnetic anisotropy landscape and yields a giant low‑field magnetostriction of 721 ppm under a field of 15.8 kA/m (19.5 mT), establishing a new route for domain engineering and achieving large, low‑field‑triggered magnetostriction.

1. Introduction

Manipulation of magnetic domain structures using non‑invasive stimuli lies at the heart of next‑generation functional materials, with broad implications for sensing and information technologies [1, 2, 3, 4]. Among various external stimuli, mechanical stress offers a particularly versatile route, enabling deterministic magnetic state switching without direct electromagnetic intervention [5, 6]. In magnetostrictive materials, stress reshapes the magnetic energy landscape via magnetoelastic coupling [7, 8, 9, 10, 11], inducing complex domain reconfigurations ranging from hierarchical branching to topological skyrmioniums [12, 13, 14, 15]. This stress‑mediated approach not only deepens our understanding of magnetomechanical coupling but also opens new pathways toward adaptive materials with dynamically tunable properties [16, 17].

A particularly attractive goal is to achieve large, low‑field‑triggered magnetostriction through nanoscale domain engineering, as nanodomain architectures dramatically reduce the energy barrier for domain reorientation [18]. Hence, a key challenge lies in designing nanoscale domain structures that yield such low‑field high‑strain behavior [18, 19, 20, 21]. Recent experimental observations of alternating zipper‑like magnetic domains within the small‑strain regions of shear bands in Fe‑Si‑B metallic glasses [22] suggest that a magnetic instability may exist in magnetoelastic systems. This possibility is supported by the known negative curvature of the effective magnetic anisotropy (EMA) energy with respect to magnetization orientation [23], which raises the prospect of a thermodynamic driving force for spontaneous domain splitting. Such a scenario is strikingly analogous to chemical spinodal decomposition in phase‑separating alloys, where a composition (c)‐dependent free energy F(c) with negative curvature ($\frac{{\partial }^{2}F}{\partial c^2}<0$) causes infinitesimal compositional fluctuations to spontaneously amplify into periodic modulations [24, 25]. Inspired by this analogy, we propose a mechanism that transforms microscale magnetic domains into alternating nanoscale patterns, namely stress‑driven ferromagnetic spinodal decomposition.

This analogy, however, raises several unresolved questions: (1) Conventional magnetization proceeds via localized nucleation and growth, involving abrupt domain‑wall displacement or local rotation [26, 27]. Can stress‑driven magnetic decomposition instead manifest as a homogeneous, continuous process characterized by global small‑angle spin deviations, mirroring compositional spinodal decomposition [28, 29, 30, 31, 32]? (2) What is the intrinsic driving force that mirrors the chemical spinodal criterion? (3) How do magnetic analogues of chemical gradient energies counteract domain decomposition?

Here, we address these questions through phase‑field simulations on a rhombohedral (R) Tb_0.3Dy_0.7Fe₂ (Tb–Dy–Fe) system with <111> easy axes. We demonstrate that a uniform microdomain (MD) structure, stabilized by [001] compressive stress, spontaneously decomposes upon gradual stress relaxation into alternating nanodomain‑in‑microdomain (NIMD) patterns with two distinct magnetization orientations. This fully reversible MD‑to‑NIMD transition is homogeneous from the outset, characterized by global, continuous small‑angle spin reorientation, and is driven by the negative curvature of the EMA energy landscape, recapitulating the thermodynamic essence of spinodal decomposition [24, 25, 33]. Remarkably, the resulting NIMD architecture delivers an ultrahigh magnetostriction of 721.5 ppm under a weak magnetic field of only 19.5 mT (along [$\overline{1}\overline{1}0$]), an 11‑fold enhancement over conventional domains. Beyond establishing stress‑driven ferromagnetic spinodal decomposition as a new paradigm for non‑equilibrium domain engineering, this work paves the way for adaptive magnetoelastic devices operating in ultralow or zero magnetic fields, as well as for real‑time strain mapping and encrypted storage technologies.

2. Results and Discussion

Figure 1 illustrates the stress‐driven magnetic domain evolution in a Tb–Dy–Fe single crystal with <111> easy axes. The domain morphology is characterized by the angle θ = arcsin(m ₃) between the magnetization vector m (m ₁, m ₂, m ₃) and the (001) plane (see the upper right corner of Figure 1). Upon applying a strong compressive stress of  σ* (σ* = 500 MPa) along [001] together with a weak magnetic field of 3 kA/m along [$\overline{1}00$], the system's magnetic moments align along the <110> directions in the (001) plane, exhibiting a uniform magnetic MD structure similar to that of the orthorhombic (O) phase in such materials [34, 35, 36]. After removing the magnetic field, gradual stress release causes this MD to split spontaneously into nanodomains with two distinct magnetic orientations, forming an alternating nanodomain‑in‑microdomain (NIMD) pattern (Figure 1a). Such small‑domain‑in‐large‐domain architectures have also been observed experimentally in Tb_0.3Dy_0.7Fe₂ [37], FeGa [38], and Fe₇₈Si₉B₁₃ [22]. Notably, the nanodomain width remains constant at 226 nm throughout the stress‐driven transformation. The resulting NIMD configurations exhibit programmable, magnetic‐field‐free memory behavior, promising applications in multistate magnetic memory and mechano‐magnetic encryption.

FIGURE 1.

Reversible MD‐to‐NIMD evolution under cyclic compressive stress in a Tb–Dy–Fe ferromagnetic system. (a) Stress‑induced MD‑to‑NIMD transition upon stress reduction, with magnetization orientation angle θ = arcsin(m ₃) relative to the (001) plane (color scale: from −46° to 46°). (b) NIMD‑to‑MD recovery upon stress application, using the same θ‑mapping protocol as in (a). (c) Spatial distribution of θ along the dashed trajectory in (a) during decreasing compressive stress. (d) Maximum θ fluctuation amplitude as a function of applied stress.

As the compressive stress decreases from 1.0 σ* to 0 σ*, the amplitude of θ increases from 0° to  ± 31.2°, completing the transition from MD to NIMD (Figure 1a). Conversely, upon gradual reloading (Figure 1b), the system exhibits a reversible NIMD‐to‐MD transition, with θ amplitude returning to 0° at 1.0 σ*. The spatial variation of θ along the dashed line in Figure 1a follows a quasi‐square wave pattern (Figure 1c), and the maximum absolute θ amplitude grows monotonically with decreasing stress, reaching 31.2° at zero applied stress (Figure 1d).

Having established the reversible MD‑to‑NIMD transition under standard conditions, we next explore how the required stress level can be reduced by tuning material parameters. Beyond the 500 MPa protocol, adjusting the λ ₁₀₀ strain magnitude from 100 to 300 ppm substantially lowers the initial compressive stress needed (Figure S1). For example, applying only 200 MPa along [001] is sufficient to drive the system into the MD state. Upon gradual stress release to 150 MPa, NIMD patterns emerge with a narrower nanodomain width of 113 nm (compared to 226 nm previously) and an increased number of quasi‑square‑wave cycles (from 4 to 8). Thus, enhancing λ ₁₀₀ significantly improves the stress sensitivity of the domain structure, enabling more efficient and lower‑stress engineering of NIMD configurations.

To resolve the dynamics of stress‑induced MD‑to‑NIMD decomposition, we monitor the temporal evolution of the domain structure and total free energy density (F_tot ) after abruptly reducing the compressive stress from σ* to 0.9 σ* at 0 t* (t* = 2.5 × 10⁻⁹ s). The time‑dependent total free energy density curve in Figure 2a reveals three distinct stages.

FIGURE 2.

Stress‐driven energy and microstructure evolution during MD‐to‐NIMD decomposition in Tb–Dy–Fe as a function of time. Here, t* = 2.5 × 10⁻⁹ s. (a) Time‑dependent total free energy density upon stress reduction from σ* to 0.9 σ*, divided into three regimes: initial (0 t*– 0.4 t*, pink), second (0.4 t* – 0.56 t*, light blue), and stabilization (> 0.56 t*, purple). (b) Emergent domain structure dynamics during initial regime (0 t* – 0.4 t*). (c) Local NIMD evolution during the period from 0.44 t* to t*. (d) Temporal evolution of θ along dashed lines I and II in the initial regime. (e) Temporal evolution of θ along the same lines for the period from 0.4 t* to t*.

The initial stage (0 t* – 0.4 t*). The total free energy density decreases only slightly. Spins along lines I and II begin to fluctuate symmetrically: those on line I tilt upward (θ > 0) while those on line II tilt downward (θ < 0). This cooperative motion leads to the uniform emergence of alternating nanodomains within the parent MDs (Figure 2b). The amplitude of θ grows progressively with time (see also Figure 1d).

The second stage (0.4 t* – 0.56 t*). The total free energy density drops sharply. Spin deviations increase rapidly across both left and right MDs (Figure 2c). The θ profiles in Figure 2e show that spins now oscillate around positive and negative half‑axes, forming sine‑wave‑like patterns.

The stabilization stage (>0.56 t ^*). The total free energy density fluctuates mildly and eventually saturates. Spins continue to deviate, ultimately adopting a quasi‑square‑wave pattern, accompanied by a progressive narrowing of domain walls.

After full decomposition, the spatial distributions of F_tot , magnetocrystalline anisotropy energy density (F_ani ), exchange energy density (F_exc ), and magnetostatic energy density (F_ms ) all exhibit NIMD‑like modulations due to strong variations at domain walls and within nanodomains (Figure S2). In contrast, the elastic energy density (F_el ) remains nearly homogeneous (Figure S2d), which starkly differs from the pronounced localization of in‑plane strain and internal stress at NIMD domain boundaries (Figure S3).

To understand the driving force behind this transition, we examine the free energy landscape. The EMA energy, which is the primary determinant of domain evolution, is formed by the elastic energy (incorporating magnetoelastic coupling [39]) together with magnetocrystalline anisotropy energy. Upon stress reduction, the system's effective easy axes immediately deviate from the (001) plane. Initially, the in‑plane magnetic moment lacks sufficient time to rotate, resulting in high EMA energy density. Over time, domains align along their respective easy axes, lowering the EMA energy density and compensating for increases in exchange, magnetostatic, and elastic energy densities (Figure S4). Consequently, the total free energy density decreases as shown in Figure 2a. Importantly, conventional magnetization proceeds via localized nucleation and growth, where the process initiates in a limited region rather than globally, involving abrupt domain‑wall motion or local rotation [26]. In contrast, the MD‑to‑NIMD transition here is characterized by global, continuous small‑angle spin deviations from the very beginning. This fundamentally different mechanism is what we term stress‑driven ferromagnetic spinodal decomposition.

To elucidate the origin of the spontaneous MD‑to‑NIMD transition, we derive the EMA energy density (F*), which combines magnetocrystalline anisotropy and magnetoelastic coupling [39]:

$$\begin{array}{c}\hfill \begin{array}{c}{F}^{\ast }=-\frac{3}{2}{\lambda }_{100}\left(C_{11}-C_{12}\right)\left[{\epsilon }_{11}{\left(\cos \theta \cos \phi \right)}^2+{\epsilon }_{22}{\left(\cos \theta \sin \phi \right)}^2\right.\hfill \\ +\left.{\epsilon }_{33}{\left(\sin \theta \right)}^2\right]-6{\lambda }_{111}{C}_{44}\left[{\epsilon }_{12}{\left(\cos \theta \right)}^2\cos \phi \sin \phi +{\epsilon }_{13}\cos \theta \cos \phi \sin \theta \right.\hfill \\ \left.+{\epsilon }_{23}\cos \theta \sin \phi \sin \theta \right]+\left(K_1+\mathrm{\Delta }{K}_1\right)\left[{\left(\cos \theta \right)}^4{\left(\cos \phi \sin \phi \right)}^2\right.\hfill \\ +\left.{\left(\cos \theta \cos \phi \sin \theta \right)}^2+{\left(\cos \theta \sin \phi \sin \theta \right)}^2\right]\hfill \end{array}\end{array}$$ (1)

where $\mathrm{\Delta }{K}_1=\frac{9}{4}(c_{11}-c_{12}){\lambda }_{100}^2-\frac{9}{2}{c}_{44}{\lambda }_{111}^2\approx -5.6\times {10}^5$ J/m ³ [26] and φ is the angle between the projection of m onto the (001) plane and the x‐axis (Figure 3a). Under [001] compressive stress, the magnetic moment rotates within planes containing the [111] diagonals, resulting in nearly equal x and y components, as seen in simulation results (Figure 3b). For analysis, we plot F*  − θ curves under various compressive stresses for the left domain with φ  ≈ 135° according to Equation (1). Figure 3c shows that at low stresses, F*  − θ curves exhibit a protrusion in the small‑angle regime, indicating negative curvature ($\frac{{\partial }^2{F}^{\ast }}{\partial {\theta }^2}<0$), as confirmed by the second derivative plot (Figure 3d). In this region, small fluctuations θ ₀ ± δθ lead to , i.e., a thermodynamically unstable condition. Consequently, the system reduces its EMA energy density through continuous, spontaneous growth of magnetization fluctuations, driving the breakup of a single MD into alternating NIMD patterns. This behavior mirrors classical spinodal decomposition, now realized in the magnetic orientation space.

FIGURE 3.

Magnetization orientation analysis and energy landscape evolution in Tb–Dy–Fe. (a) Definition of out‑of‑plane angle θ and in‑plane angle φ in 3D coordinates. (b) Spatial distribution of magnetic moment on the (001) plane. (c) EMA energy density as a function of θ under different compressive stresses. (d) Second derivative of EMA energy density, highlighting stress‑modulated spinodal instability regions.

Notably, decreasing the compressive stress broadens the angular range where $\frac{{\partial }^2{F}^{\ast }}{\partial {\theta }^2}<0$ (Figure 3d), enlarging the spinodal window and increasing the final θ amplitude (Figure 1d). At zero stress, the EMA energy density minima occur at θ  ≈ ± 31.2° (Figure 3c), precisely matching the observed θ amplitude of the alternating R‑type NIMD structure.

Our simulations also reveal a subtle interplay of competing energetic factors. Even when the applied stress is increased back to σ* (merging NIMD into a single MD), the upward curvature in the EMA energy density persists up to 3 σ* (Figure 3c). This hysteresis, where gradual unloading from 3 σ* does not immediately trigger decomposition, points to an inherent limitation of the current energy formulation: the absence of explicit resistance terms (e.g., domain‑wall pinning or gradient‑energy analogues) that oppose domain splitting. This observation motivates future refinements to fully capture the kinetic asymmetry of ferromagnetic spinodal decomposition.

To directly probe the competing energetic factors that govern the decomposition, we perform controlled simulations by slightly adjusting the magnetic exchange coefficient and magnetocrystalline anisotropy constant. Increasing the exchange coefficient from 0.80 A to 1.20 A (where A  = 0.9 × 10⁻¹¹ J/m), while keeping all other parameters constant, progressively raises the critical stress reduction needed for the apparent MD‑to‑NIMD transition (from 0.98 σ* to 0.96 σ*; Figure 4a). This is because the magnetic exchange interaction, which favors parallel alignment of magnetic moments [26], acts as a stabilizing force against domain decomposition. This behavior mirrors the role of chemical gradient energy in classical spinodal decomposition [33]. In contrast, strengthening the anisotropy constant from 0.98 K ₁ to 1.02 K ₁ (K ₁ = − 0.6 × 10⁵ J/m³) while all other parameters remain unchanged lowers the required stress reduction for the pronounced decomposition (from 0.97 σ* to 0.99 σ*; Figure 4b). Enhanced anisotropy promotes preferential alignment along the <111> easy axes, generating sufficient magnetic moment fluctuations to overcome resistive terms such as magnetic exchange coupling. In other words, magnetocrystalline anisotropy serves as the primary driving force for domain splitting.

FIGURE 4.

Competing energy terms governing stress‑mediated domain evolution pathways in Tb–Dy–Fe. (a) Exchange‑stiffness dependence of the MD‑to‑NIMD transition during stress relaxation (0.80 A, 1.00 A, and 1.20 A), with all other parameters unchanged. (b) Magnetocrystalline anisotropy modulation under stress removal (0.98 K ₁, 1.00 K ₁, and 1.02 K ₁), with all other parameters unchanged. (c) Rapid stress release (0.25 σ* per step) yields a metastable uniform spin‑tilted state, bypassing equilibrium alternating‑domain formation.

Thus, stress‐driven ferromagnetic spinodal decomposition represents a novel phase transition occurring near metastability boundaries. Near the spinodal regime, a delicate balance emerges between driving forces (e.g., magnetocrystalline anisotropy) and resistive mechanisms (e.g., magnetic exchange coupling). As the system approaches this critical threshold, amplified magnetization fluctuations spontaneously drive the formation of alternating NIMD patterns. Remarkably, the final morphology is governed by kinetic pathways: gradual stress relaxation yields equilibrium NIMD patterns, whereas rapid stress release (0.25 σ* per step) produces a uniform spin‐tilted metastable state that entirely bypasses domain alternation (Figure 4c). This kinetic bifurcation underscores the intrinsically non‐equilibrium nature of ferromagnetic spinodal decomposition.

To evaluate the low‑field magnetostrictive performance, we compare three distinct domain states: ND (normal domain, without compressive stress treatment), SRMD (stress‑released microdomain, obtained by applying and then fully releasing σ*), and NIMD (formed by incremental stress reduction of 10 MPa per step to 0 MPa; Figure 5a). Their low‑field strain response follows a clear hierarchy: NIMD > SRMD > ND (Figure 5b). At a magnetic field of 15.8 kA/m (equivalent to μ ₀ H = 19.5 mT, where μ ₀ is the vacuum permeability), the NIMD structure delivers a magnetostrain of 721.5 ppm, which is 11 times higher than that of NDs.

FIGURE 5.

Field‑driven domain evolution, magnetostrain performance and energy density responses in Tb–Dy–Fe. (a) Domain evolution under increasing magnetic field for ND, SRMD, and NIMD phases. (b) Low‑field magnetostrictive response of the three phases under a [$\overline{1}\overline{1}0$] magnetic field. At 15.8 kA/m (μ ₀ H = 19.5 mT), the NIMD structure delivers a magnetostrain of 721.5 ppm. (c) Comparison of magnetostriction vs. magnetic field for NIMD‑structured Tb–Dy–Fe with other typical magnetostrictive alloys. (d) Schematic of magnetic moment rotation induced by the applied field. (e) External magnetic field energy density evolution for ND, SRMD, and NIMD phases; inset: magnetostatic energy density variations.

Figure 5c benchmarks the NIMD performance against several representative magnetostrictive materials: directionally solidified (DS) Fe_67.7Pd_32.3 alloy [18], rare‐earth‐iron alloys (e.g., Terfenol‐D) [40, 41], single‐crystal (SC) ferromagnetic shape memory alloys (FSMAs) [42, 43], and Fe–Ga [44]. Rare‑earth‑iron alloys require an applied field of μ ₀ H ≈ 1 T to achieve comparable strain, whereas even the best low‑field material among these (DS Fe_67.7Pd_32.3) needs an 80 mT field, which is four times higher than that needed for our NIMD structure [18]. These results underscore the substantial advantage of the stress‑engineered NIMD architecture, consistent with previous reports that strategically designed nanodomains amplify low‑field strain responses in ferroic systems [18, 19, 20, 21].

The rhombohedral Tb–Dy–Fe phase has <111> easy axes. Under a weak [$\overline{1}\overline{1}0$] magnetic field, the NIMD structure remains stable in the [$\overline{1}\overline{1}1$] (red) and [$\overline{1}\overline{1}\overline{1}$] (light blue) domains, whose magnetic moments are nearly aligned with the field (Figure 5d). In contrast, the [$\overline{1}1\overline{1}$] (yellow) and [$\overline{1}11$] (black) domains exhibit larger misalignment angles and undergo domain‑wall‑displacement‑driven reconfiguration toward the energetically favorable [$\overline{1}\overline{1}1$] and [$\overline{1}\overline{1}\overline{1}$] orientations, reducing Zeeman energy density. When the field reaches 15.8 kA/m, the system transforms completely into an alternating domain configuration (Figure 5a). This transformation not only lowers the external field energy density but also reduces the demagnetization energy density (Figure 5e). By contrast, SRMDs show only minor structural changes under the same field (slight contraction of black domains and expansion of light blue ones), while NDs with random orientations remain largely unaffected in the 0–20 kA/m range.

The critical transition at 15.8 kA/m is accompanied by coordinated energy density jumps: a discontinuity in total energy density aligns with the external field energy density trend, along with abrupt reorganizations in magnetocrystalline anisotropy energy density and characteristic reductions in elastic and exchange energy densities (Figure S5). These coupled energy density pathways explain the superior weak‑field responsiveness of the NIMD configuration. In essence, the NIMD structure consists of two sets of alternating R‑type nanodomains. Those with larger angles relative to the [$\overline{1}\overline{1}0$] field can convert via domain‑wall displacement into the set with smaller angles, rapidly reducing the total free energy density and amplifying lattice deformation, leading to the observed giant low‑field magnetostriction.

It is worth noting that the ferromagnetic morphotropic phase boundary (MPB) features two nanodomains of distinct phase structures coexisting within a single micron‑sized domain, also providing a low energy landscape for large magnetostriction [37, 45, 46]. Intrinsically governed by composition and temperature, the MPB represents a thermodynamic two‑phase coexistence. In contrast to this composition/temperature controlled MPB, the present stress‑driven spinodal switching operates within a single rhombohedral phase, without any compositional or structural phase transition. This process initiates with global, continuous small‑angle spin deviations, proceeds homogeneously, and is fully reversible. Its origin lies in the negative curvature of the EMA energy landscape, mirroring spinodal instabilities in alloy systems. Moreover, unlike the magnetic‑field‑driven spinodal magnetization switching mechanism [47], the present stress‑driven transformation between MD and NIMD follows an entirely different pathway, underscoring the distinct physical origin of our approach. Overall, these findings establish a fundamentally new route to create alternating nanodomain patterns by stress engineering, offering a novel strategy for achieving large magnetostriction under low magnetic fields.

3. Conclusion

Using phase‑field simulations, we demonstrate a reversible, stress‑driven transition between magnetic domain configurations in a rhombohedral Tb–Dy–Fe alloy with <111> easy axes. Gradual unloading of [001] compressive stress spontaneously transforms a uniform MD into alternating NIMD patterns. This transition occurs within the same rhombohedral phase, entailing only spatial redistribution of magnetic domains. It initiates with global, continuous small‑angle spin deviations and proceeds homogeneously, rather than via localized nucleation and growth that involves abrupt domain‑wall motion or local rotation. The underlying mechanism is the negative curvature of the EMA energy density landscape, a magnetic analogue of chemical spinodal decomposition in materials. Moreover, magnetic exchange interactions are found to oppose the decomposition, whereas magnetocrystalline anisotropy serves as the primary driving force.

The resulting NIMD architecture delivers exceptional low‑field magnetostrictive performance: a strain of 721.5 ppm under a [$\overline{1}\overline{1}0$] magnetic field of only 19.5 mT, which is 11 times higher than that of normal domains. This giant response originates from domain‑wall displacement that reorients R‑type nanodomains from larger to smaller angles relative to the field, rapidly reducing free energy density and amplifying lattice deformation.

Our findings establish stress‑driven ferromagnetic spinodal decomposition as a new paradigm for non‑equilibrium domain engineering. The resulting large magnetostriction under low magnetic fields makes this approach promising for ultralow‑field magnetoelastic devices. Furthermore, the stress‑sensitive periodicity of NIMD patterns enables direct mapping of localized strain fields via magnetic imaging, offering promising routes for encrypted storage and stress visualization. Future work may extend this strategy to other materials and stress states to optimize the low‑field magnetostrictive response for practical applications.

4. Experimental Section

In the phase‐field micromagnetic microelastic model, the domain evolution is governed by Landau‐Lifshitz‐Gilbert (LLG) equation [39],

$$\frac{\partial m}{\partial t}=-\frac{{\gamma }_0}{1+{\alpha }^2}\left[m\times H_{eff}+\alpha m\times \left(m\times H_{eff}\right)\right]$$ (2)

where m is the unite magnetization vector; t is the time; γ ₀ is the gyromagnetic ratio; α is the damping constant. The effective magnetic field H_eff in Equation (2) is determined by the derivative of the total energy density F_tot against magnetization. Here, F_tot is the sum of the magnetocrystalline anisotropy energy density F_ani , the magnetic exchange energy density F_exc , the magnetostatic stary field energy density F_ms , the external magnetic field energy density F_ext , and the total elastic energy density F_el , that is, F_tot = F_ani   + F_exc + F_ms + F_ext + F_el [39]. The total elastic energy density F_el is given by

$$F_{el}=\underset{V}{\int }\frac{1}{2}{c}_{ijkl}({\epsilon }_{ij}-{\epsilon }_{ij}^0)({\epsilon }_{kl}-{\epsilon }_{kl}^0)dV-V{\sigma }_{ij}{\overline{\epsilon }}_{ij}$$ (3)

where c_ijkl is the elastic stiffness tensor, with summation over repeated indices (i, j, k, l = 1, 2, 3); ${\epsilon }_{ij}^0$ is the stress‐free strain coupled to the local magnetization through magnetostriction; V is the total volume; σ _ij is the applying stress. The total strain ε _ij in Equation (3) is written as ${\epsilon }_{ij}={\overline{\epsilon }}_{ij}+{\epsilon }_{ij}^{het}$ with the stress‐driven homogeneous strain ${\overline{\epsilon }}_{ij}$ and the spatially dependent heterogeneous strain ${\epsilon }_{ij}^{het}$[39, 48]. Here, $\int {\epsilon }_{ij}^{het}dV=0$ and ${\epsilon }_{ij}^{het}=\frac{1}{2}(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})$, where the displacement u_i in the mechanical equilibrium equation is solved using Fourier spectral iterative perturbation method [39, 48].

The materials parameters for the R‐type Tb–Dy–Fe are taken from Ref. [36, 48]. The simulations are performed using magnetocrystalline anisotropy constants K ₁ = − 0.6 × 10⁵ J/m³ and K ₂ = − 2 × 10⁵ J/m³, which correspond to a temperature of T = 320 K. The other material constants are: elastic constants c ₁₁ = 1.41 × 10¹¹ N/m², c ₁₂ = 6.48 × 10¹⁰ N/m², c ₁₄ = 4.87 × 10¹⁰ N/m²; magnetostrictive constants λ ₁₁₁ = 1600 ppm, λ ₁₀₀ = 100 ppm; saturation magnetization M_s = 0.8 × 10⁶ A/m; exchange stiffness A  = 0.9 × 10⁻¹¹ J/m; and damping constant α  = 0.1. The geometry is discretized in finite difference cells of size 512 × 512 × 1 with a grid spacing of Δx  = 5 nm and periodic boundary conditions. The time‐dependent LLG equation is solved using the Gauss‐Seidel projection and the semi‐implicit Fourier spectral method [39, 48]. To initialize the simulation, a stable MD structure is first generated by applying a [001] homogeneous compressive stress  σ* = 500 MPa together with a weak magnetic field of 3 kA/m along the [$\overline{1}00$] direction. After completely removing the magnetic field, the stress is gradually decreased or increased to obtain the evolution of the stress‐driven magnetic domain structure. For the magnetostriction curve simulations, the external magnetic field is increased stepwise by 0.1 A/m per step, and data are sampled every 200 steps.

Author Contributions

Zhengming Zhang conceived the idea, led the overall study and wrote the manuscript with inputs from Hongchang Wang. Hongchang Wang and Zhengming Zhang developed the simulation code, performed the simulations, acquired the data, and generated the data visualizations. Zhengming Zhang and Hongchang Wang analyzed the results, proposed the theoretical model, and carried out the validation. Zhengming Zhang and Dunhui Wang supervised the project. Dunhui Wang and Jianhu Gong contributed to technical discussions and commented on the manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Supporting File: smll74544‐sup‐0001‐SuppMat.docx.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.