Large and sensitive magnetostriction in ferromagnetic composites with nanodispersive precipitates

Large and sensitive magnetostriction (large strain induced by small magnetic fields) is highly desired for applications of magnetostrictive materials. However, it is difficult to simultaneously improve magnetostriction and reduce the switching field because magnetostriction and the switching field are both proportional to the magnetocrystalline anisotropy. To solve this fundamental challenge, we report that introducing tetragonal nanoprecipitates into a cubic matrix can facilitate large and sensitive magnetostriction even in random polycrystals. As exhibited in a proof-of-principle reference, Fe–Ga alloys, the figure of merit—defined by the saturation magnetostriction over the magnetocrystalline anisotropy constant—can be enhanced by over 5-fold through optimum aging of the solution-treated precursor. On the one hand, the aging-induced nanodispersive face-centered tetragonal (FCT) precipitates create local tetragonal distortion of the body-centered cubic (BCC) matrix, substantially enhancing the saturation magnetostriction to be comparable to that of single crystal materials. On the other hand, these precipitates randomly couple with the matrix at the nanoscale, resulting in the collapse of net magnetocrystalline anisotropy. Our findings not only provide a simple and feasible approach to enhance the magnetostriction performance of random polycrystalline ferromagnets but also provide important insights toward understanding the mechanism of heterogeneous magnetostriction. The performance of materials that convert magnetic to mechanical energy and vice versa has been improved by introducing nanoprecipitates into their structure. Magnetostrictive materials can expand or contract under the influence of an external magnetic field. This property makes them useful for low-power sensors and energy harvesting. Simultaneously optimizing both the size of the magnetostrictive effect and the material’s sensitivity to small magnetic fields is difficult because increasing one effect tends to decrease the other. Tianyu Ma, Xi’an Jiaotong University, China, and Xiaobing Ren, National Institute for Materials Science, Tsukuba, Japan, and co-workers developed an approach to solve this problem by introducing tiny imperfections into a ferromagnetic alloy. By changing the atomic structure of small regions within the alloy iron−gallium, the researchers were able to achieve a five-fold improvement in the material’s magnetostrictive performance. a Aging-time dependent saturation magnetostriction for Fe73Ga27 random polycrystals. Bright-field images for the b 1373 K-quenched, c 1 h-aged and d 12 h-aged Fe73Ga27 random polycrystals. e Figure of merit as a function of aging time for Fe81Ga19 random polycrystals. f Comparison of saturation magnetostriction among our optimally aged Fe–Ga random polycrystals, the quenched random polycrystals doped with a third element and the single crystals.


Introduction
Magnetostrictive materials that can convert magnetic energy into mechanical energy or vice versa have been widely used in actuators, transducers, sensors, and energy harvesters [1][2][3] . For most applications, it is desirable to have a combination of large strain and a small switching field (which can be evaluated by the critical field, at which d 33 reaches a maximum 4 , and is proportional to K/M s for random polycrystals, where K is the magnetocrystalline anisotropy constant and M s is the saturation magnetization 5 ). However, it is difficult to simultaneously obtain a large magnetostriction and small switching field in a homogeneous ferromagnet because both the magnetostriction and switching fields are proportional to the magnetocrystalline anisotropy (MCA) 1 . There is one exception: combining two terminals with large magnetostriction but opposite MCA constants can yield large magnetostriction with reduced switching fields at a specific composition/ temperature, as found in the well-known pseudobinary intermetallic compounds RR′Fe 2 (R and R′ refer to two rare earth elements) 2,6-8 . When maximizing the magnetostriction for a given material, there is still a trade-off between a large magnetostriction and a small switching field, analogous to the strength-ductility trade-off in structural materials. The usual approach to maximize the magnetostriction of a given material with a large spontaneous magnetostriction constant has been fabricating a single crystal and further applying prestress or performing postgrowth magnetic annealing [9][10][11] . This complex process can constrain the ferromagnetic domains within the plane perpendicular to the crystal axis and can yield a large magnetostriction up to the theoretical limit through pure 90°domain switches when applying a magnetic field along the crystal axis. For decades, this approach has dominated the fabrication and application of the state-of-the-art magnetostrictive material Terfenol-D (e.g., Tb 0.3 Dy 0.7 Fe 2 ) with a large rhombohedral spontaneous magnetostriction constant λ 111 of 1640 ppm at room temperature 2 . The obtained technical strain λ //s along the field direction can be as high as 2400 ppm (nearly equivalent to the theoretical limit 3/2λ 111 ) by applying compressive prestress on the single crystal 11,12 . However, a large magnetostriction is always obtained at the expense of an enlarged switching field (i.e., deteriorated sensitivity) because prestress and magnetic annealing inevitably induce additional anisotropy. Consequently, it is highly desirable to find a feasible approach that can enhance the magnetostriction and reduce the switching field simultaneously.
Several theoretical studies separately note that the exchange of nonequal crystallographic axes of the tetragonal strain nanodomains embedded within a cubic matrix will output large magnetostriction 13 , and the random magnetic coupling between them will result in the collapse of net MCA 14,15 . Experimentally, it is feasible to obtain such a nanostructured composite by aging a cubic supersaturated solid solution to form nanodispersive tetragonal precipitates, as extensively reported in structural alloys. In this work, we obtained ferromagnetic composites by isothermal aging of the metastable cubic precursor to precipitate nanodispersive tetragonal particles. Our results reveal that this can be a simple and feasible approach to obtain the desired large and sensitive magnetostriction in random polycrystals, as exhibited by the proof-of-principle reference Fe-Ga alloys.
The selection of Fe-Ga alloys takes into account the large magnetostriction at small fields and, in particular, the apparent structural differences between metastable and equilibrium states. In fact, Fe-Ga alloys have stimulated extensive research efforts since their large magnetostriction was discovered in 2000 16 . As shown in two review papers 17,18 , previous research focused mainly on the compositions in the vicinity of 19 and 27 Ga at.%, at which two magnetostriction peaks have been reported 19 . The obtained λ //s of these two alloys has been successfully enhanced from~60 ppm for random polycrystals 16 tõ 200 ppm for [001]-oriented single crystals or polycrystals 20,21 and further to above 300 ppm (near the theoretical limit 3/2λ 100 ) by applying a coaxial compressive prestress or by performing postgrowth magnetic/stress annealing 9,10,21,22 . More importantly, these two alloys bear an average body-centered cubic (BCC) structure in the quenched state but have biphase A2 + L1 2 (ordered facecentered cubic) for Fe 81 Ga 19 and single L1 2 for Fe 73 Ga 27 in the equilibrium state (see the phase diagram in Fig. S1) 23 . Since the L1 2 phase has negative magnetostriction (in contrast to the A2, B2 and D0 3 phases) [24][25][26][27] , approaching equilibrium conditions has been undesirable for achieving large positive magnetostriction. However, as shown hereinafter, short-term aging of the BCC precursor induces L6 0 -type face-centered tetragonal (FCT) nanoprecipitates instead of the equilibrium L1 2 phase. The unique nanostructure with nanodispersive FCT precipitates embedded in the BCC matrix developed by optimum aging can create fivefold enhancement in the figure of merit due to the simultaneous magnetostriction enhancement and the collapse of net MCA. The obtained λ //s in the optimally aged Fe 81 Ga 19 random polycrystal can reach as high as 200 ppm, which is comparable to that for single crystal materials measured under stress-free conditions 20,21 .

Materials and methods
Ingots with nominal compositions of Fe 81 Ga 19 and Fe 73 Ga 27 were prepared by induction melting Fe and Ga with purities of 99.99% in an argon atmosphere. Cylindrical rods were prepared by subsequent casting into a steel mold with an inner diameter of 12 mm. The rods were sealed in quartz tubes filled with high-purity argon and subjected to heat treatment. The metastable BCC precursors were obtained by annealing the rods for 72 h at 1373 K, followed by quenching in ice water. An electron back-scattered diffraction (EBSD, HITACHI-SU3500) image of the 1373 K-quenched sample revealed randomly oriented grains of several hundred microns (Fig. S2). The Fe 73 Ga 27 and Fe 81 Ga 19 rods were then cut into different sections (Ø 12 × 20 mm 3 ) and aged at 723 K or 673 K, followed by quenching in ice water.
Note that during aging, the Fe-Ga alloys undergo a diffusion-controlled phase transformation [25][26][27] , after which the microstructure has a strong surface effect. To avoid the surface effect, sheets with dimensions of 10 × 3 × 1.5 mm 3 were cut from the center sections of the aged rods (the length direction was parallel to the rod axis) and subjected to the following magnetostriction measurements and microstructural characterizations. Room temperature initial magnetization curves were measured using a SQUID magnetometer, which allows us to calculate the MCA constant K 1 and saturation magnetization M s according to the law of approaching saturation. Magnetostriction along the magnetic field direction was measured at room temperature using the strain gauge method. The strain gauge had a length of 1 mm and a width of 1.5 mm (KYOWA, KFR-1-120-C1-11), which was much larger than the grain size of several hundred microns for the solution-treated sample (Fig. S2) and several microns for the overaged samples. To avoid possible stress-induced impurities, the sample surface was carefully polished by standard mechanical grinding and electropolishing methods. At least five magnetization cycles (one cycle refers to increasing the magnetic field from zero to a designed strength, decreasing it to zero, then increasing the opposite magnetic field to the same designed strength, and finally decreasing it to zero) were measured for each sample. In addition, a random polycrystalline Fe 72 Ga 28 alloy was selected as another reference to reproduce the aging effect on magnetostriction properties. Nanoindentation (G200) and atomic force microscopy (AFM) were conducted on the polished surfaces to investigate the modulus difference between the nanoprecipitates and the matrix. The foils for TEM characterization were prepared by twin-jet electropolishing at temperatures below 243 K. TEM characterizations were performed using a JEOL JEM-2100F microscope at 200 kV. The high-resolution TEM (HRTEM) images and geometric phase analysis (GPA) were analyzed using DigitalMicrograph (Gatan). Magnetic domain structures were observed under the Fresnel Lorentz mode. An inplane magnetic field was applied to observe the domain structure evolution. The optimally aged Fe 73 G 27 sample (cut into sheets with dimensions of 10 × 10 × 1.5 mm 3 ) was subjected to in situ small-angle neutron scattering (SANS) measurements using unpolarized neutrons with a wavelength of 5.3 Å, which was performed on the smallangle neutron spectrometer at the China Mianyang Research Reactor (CMRR). An in-plane magnetic field was applied along the length direction.

Results and discussion
Aging-time-dependent magnetostriction and microstructure Figures 1 and 2 show the magnetostriction and magnetization curves of the Fe 73 Ga 27 and Fe 81 Ga 19 samples, respectively. Both alloys exhibit similar aging-timedependent magnetostrictions. In the solution-treated state, the saturation magnetostriction along the field direction (λ //s ) for both alloys is 50-65 ppm (varies with composition), which is comparable to the values for Fe-Ga random polycrystals reported by Clark et al. 16 . After isothermal aging, the magnetostriction first increases to a peak value and then decreases to nearly zero or negative values with increasing aging time. Figure 1a shows the aging-time-dependent λ //s for Fe 73 Ga 27 random polycrystals, where λ //s increases significantly from 65 ppm for the 1373 K-quenched state to 172 ppm for the one aged for 1 h at 723 K but gradually decreases to −75 ppm for the one aged for 12 h. Representative magnetostriction curves are shown in Fig. 1b. The highest magnetostriction λ //s for the Fe 81 Ga 19 alloy is 202 ppm after aging for 8 h at 673 K,~4 times that for the solutiontreated state (Fig. 2). Further magnetostriction measurements on the Fe 72 Ga 28 random polycrystalline samples reproduce this aging time dependence (Fig. S3). More importantly, the obtained 202 ppm magnetostriction is comparable to that for a single crystal measured under stress-free conditions 20,21 and surpasses the reported values in bulk Fe-Ga random polycrystals doped with a third element, such as Cr, Mn, Co, Dy, or Tb 19,28-31 , as shown in Fig. 2d. Therefore, short-term isothermal aging is a simple and feasible approach to achieve large magnetostriction in Fe-Ga random polycrystals without doping with any expensive and scarce metals or the need to grow single crystals on a special instrument.
Isothermal aging also results in the collapse of the net MCA, thereby significantly enhancing the figure of merit, as shown in Fig. 2c. The MCA constant K 1 calculated according to the law of approaching saturation exhibits a nonlinear dependence on aging time. K 1 for the 1 h-aged Fe 73 Ga 27 sample and the 8 h-aged Fe 81 Ga 19 sample is even smaller than that for the 1373 K-quenched sample, which is unusual for a homogeneous ferromagnet since a larger magnetostriction should result from strengthened MCA 1 . The reduced K 1 (collapse of net MCA) indicates that the alloys become much softer after optimum aging. The figure of merit can be defined by │λ //s /H s │ 32 or │λ //s / K 1 │ 33 or │λ //s /H c │ 7 , where H s is the magnetostriction saturation field and H c is the coercivity. Since both H s and H c are proportional to │K 1 │for random polycrystalline ferromagnets, here, we use │λ //s /K 1 │to evaluate the figure of merit of the Fe-Ga random polycrystals. The │λ //s /K 1 │ for the optimally aged Fe 81 Ga 19 sample is nearly 5 times larger than that of the quenched state, as shown in Fig. 2c. This means that after optimum aging, the magnetostriction becomes more sensitive, i.e., it is very easy to achieve large and sensitive magnetostriction. In addition, the 1 haged Fe 73 Ga 27 sample exhibits only 3.2% higher saturation magnetization M s than the 1373 K-quenched sample ( Fig. 1b) but 186.7% larger λ //s . This is also against conventional wisdom for homogeneous ferromagnets because │λ s │ is proportional to M s 31 . Figure 3 shows bright-field images and the corresponding selected area electron diffraction (SAED) patterns of the Fe 73 Ga 27 samples with different heat treatments. The superlattice reflections at the {200}* positions suggest that the BCC phase is ordered at the asquenched state (Fig. 3d). According to previous studies 26,34 , this state contains a relatively large fraction of the D0 3 phase in addition to the disordered A2 phase. After aging for 1 h at 723 K, randomly distributed nanoprecipitates are formed (Fig. 3b)  the L6 0 -type FCT structure is alternatively a bodycentered-tetragonal (BCT)-modified D0 3 structure, as suggested by Lograsso et al. 35 . The obtained FFT patterns are consistent with the simulated patterns for the modified D0 3 structure, as reported by Jiang et al. [36][37][38] . L6 0type nanoprecipitates are also observed in the 8 h-aged Fe 81 Ga 19 alloy (Fig. S4). An important conclusion can then be drawn: the shortterm aging-induced substantial magnetostriction enhancement is attributed to the formation of L6 0 -type nanoprecipitates, and the overaging-induced magnetostriction deterioration is due to the further transformation into the L1 2 phase with intrinsic negative magnetostriction. It should be noted that, in the past, the quenched state has been believed to exhibit the largest magnetostriction, and approaching the equilibrium state (e.g., slow cooling or increasing aging time) will deteriorate the magnetostriction [39][40][41] . Numerous investigations have shown that slow cooling will promote the ordering of the D0 3 phase and/or the precipitation of the L1 2 /D0 19 phases, which are detrimental to magnetostriction performance 34,[42][43][44] . For instance, A.K. Mohamed et al. recently showed that the A2 → D0 3 + A2′ transition will occur above 673 K with cooling rates of 2-30 K/min and that the L1 2 phase will appear with cooling rates ≤2 K/ min in Fe 80.5 Ga 19.5 alloys 43 . On the other hand, long-term aging within the L1 2 phase regime will lead to a large fraction of the L1 2 phase and deteriorate magnetostriction 24,27,45 . For instance, the magnetostriction of the Fe 72.4 Ga 27.6 alloy decreases from 105 to 30 ppm when the 50 vol.% L1 2 phase is formed 27 . However, the present work reveals that placing Fe-Ga alloys at a proper intermediate state between the metastable and equilibrium states (e.g., short-term aging) should be the optimum condition to maximize magnetostriction, owing to the formation of dispersive L6 0 nanoprecipitates. The role of L6 0 nanoprecipitates in enhancing magnetostriction will be discussed in the next section.

Role of FCT nanoprecipitates
Note that the formation of the L1 2 26,46 ; thus, it has slow kinetics. Therefore, the transformation process and the product phase are largely affected by the competition between the strain energy and interfacial energy. In the early transformation stage, given that the atomic positions within the nanoprecipitates have been altered to fit the L1 2 lattice, the lattice distortion will be constrained by the surrounding matrix because of the limited size. As shown in Fig. 3, the extra superlattice reflections of the L6 0 nanoprecipitates indicate that the atomic position exchanges occur at local regions; however, the lattice parameters are not fixed for the equilibrium L1 2 . For instance, the calculated tetragonality (c/a) for the FCT L6 0 nanoprecipitates is 0.712 in Fig. 4e, h, which is slightly greater than 1/ ffiffi ffi 2 p for an ideal BCC structure and much smaller than 1 for an ideal FCC structure. In this case, the L6 0 nanoprecipitates can be deemed the constrained-L1 2 phase. The unit cells of D0 3 , L6 0 and L1 2 Fe 3 Ga are shown in Fig. S5. Very recently, our work indicated that L6 0 nanoprecipitates have size-dependent tetragonality; the larger their size is, the greater the tetragonality (closer to 1 of the equilibrium L1 2 phase) 47 .
It should be noted that the BCC matrix has an open structure and that the L6 0 and L1 2 phases are more closely packed. The nanoindentation test revealed that the 1 haged sample yields the precipitation hardening effect due to the formation of L6 0 nanoprecipitates, which are mechanically harder than the BCC matrix (Fig. S6). The mutual elastic interaction between the harder L6 0 nanoprecipitates and the softer BCC phase locally distorts the surrounding matrix, although the formation of L6 0 multivariants reduces the net strain. One can see from the diffraction pattern in Fig. 3e that diffuse scattering of the fundamental reflections indicates local lattice distortion. More clearly, the lattice distortion is reflected by the   Fig. 3d, f, respectively. Figure 4c shows a typical distorted region surrounding the L6 0 nanoprecipitate, which is an inverse FFT image created by taking the (040), 040 À Á , (400), and 400 À Á reflections. Local strain fluctuations can be seen from the inequivalent d spacings and between the (010) and (100) planes. Such lattice distortion is also observed along the [111]-and [110]-D0 3 zone axes, as shown in Fig. 4f, i, respectively. The local lattice distortion surrounding the nanoprecipitates is also verified by geometric phase analysis (Fig. S7). The local tetragonal distortion then gives rise to an apparent average tetragonality greater than 1 [splitting of the (12 00)-D0 3 reflection in the bottom of Fig. 3e], which means that the spontaneous magnetostriction constant λ 001 of the matrix is effectively increased after short-term aging [compared with the (12 00)-D0 3 reflection without splitting in the bottom of Fig. 3d].  Fig. 5e). Such DWs are due to the domain wall energy density difference between the matrix and L6 0 nanoprecipitates. Unlike the straight and well-ordered samples for the 1373 K-quenched sample, after aging, the DWs transform into serpentine and irregularly shaped configurations. This indicates that the L6 0 nanoprecipitates are magnetically coupled with the matrix phase. The irregular magnetic domain configurations of the matrix are the result of the collapse of net anisotropy, which stems from the competition between the coherent anisotropy and the random anisotropy of the L6 0 nanoprecipitates 15 . Figure 5g-i show in situ L-TEM images for the 1 h-aged Fe 73 Ga 27 sample. When a small magnetic field of 0.006 T (60 Oe) is applied, the magnetic DWs of some L6 0 nanoprecipitates disappear and the ferromagnetic domains of the matrix simultaneously rotate toward the field direction. When the magnetic field increases to 0.015 T (150 Oe), the DWs of the matrix disappear, whereas those of several L6 0 nanoprecipitates still exist. This indicates that the L6 0 nanoprecipitates with easy axes far away from the field direction are certainly more difficult to magnetize than those with easy axes close to the field direction.
We further performed in situ SANS measurements on the 1 h-aged Fe 73 Ga 27 sample. Figure 6a-d show the original 2D SANS patterns detected in different magnetic fields. The contour of 2D patterns undergoes an isotropyanisotropy-isotropy transition with increasing magnetic field. Figure 6e, f show the scattering intensity I(q) curves summed over the scattering vectors q x (parallel to H) and q y (perpendicular to H) from 2D patterns, respectively. In the I-q x curves (Fig. 6e), a broad peak is detected in the range of 0.1 nm −1 ≤ q x ≤ 0.3 nm −1 at 0 T, 0.05 T and 0.15 T; however, this peak disappears when the field increases to 0.5 T or 1.7 T. In the I-q y curves (Fig. 6f), a broad peak is also observed at 0 T and 0.05 T, whereas this peak disappears at fields above 0.15 T. This magneticfield-dependent broad peak corresponds to inhomogeneous L6 0 nanoprecipitates. In the nanoprecipitate-matrix approximation 48,49 , the scattering intensity I(q) reflects the spatial fluctuations of scattering length density (SLD) for both nuclear and magnetic contributions. The nuclear contribution can be expressed as NV 2 p FðqÞ j j 2 Δρ 2 nuc , and the magnetic contribution can be expressed as NV 2 p FðqÞ j j 2 Δρ 2 mag sin 2 α, where N stands for the number of nanoprecipitates; V p is the volume of an individual nanoprecipitate with the form factor F(q); Δρ nuc and Δρ mag are the contrasts of nuclear and magnetic SLDs between the nanoprecipitates and the matrix, respectively; and α is the angle between q and the local magnetization vector in the nanoprecipitates.
The calculated nuclear SLD is~7.2 × 10 −6 Å −2 for both the BCC matrix and the L6 0 nanoprecipitates (Fig. S8), which is consistent with the value reported in the literature 50 . This indicates that there is no measurable nuclear contrast between the L6 0 nanoprecipitate and matrix. In addition, since the L6 0 nanoprecipitates formed by diffusive phase transformation cannot be decomposed by increasing the magnetic field, the magnetic-fielddependent broad peak is not attributed to their nuclear contrast but rather to the magnetic contrast between the L6 0 nanoprecipitates and the matrix. Under 1.7 T (at which the magnetization is saturated), all the magnetizations are parallel to the applied field direction, and the I-q y curve can then reflect the magnetic contribution due to the contrast of magnetic SLDs between the L6 0 nanoprecipitates and matrix. According to the fitting curve (Fig. S8a), the calculated average radius for the spherical nanoprecipitates is 9.7 ± 0.1 nm (consistent with the TEM observations) and the calculated Δρ mag (the difference in magnetic SLDs between the L6 0 nanoprecipitates and matrix) is~4.0 × 10 −7 Å. We further calculated the magnetic SLD for the BCC matrix and the L6 0 nanoprecipitates, which is 2.9 × 10 −6 Å −2 and 3.3 × 10 −6 Å −2 , respectively.
The distinct feature between the I-q x and I-q y curves at 0.15 T can correlate the magnetization flips of the L6 0 nanoprecipitates with the magnetostriction since the magnetostriction is nearly saturated in this field (Fig. 1b).
The magnetization process of the 1 h-aged Fe 73 Ga 27 sample can be derived from the in situ characterizations, as schematically illustrated in Fig. 6g-i. For ease of understanding, the nanoprecipitates with the c-axis parallel (or close) and perpendicular to (or far away from) H are defined as L6 0// and L6 0⊥ , respectively. According to the observations of the DWs surrounding the L6 0 nanoprecipitates in Fig. 5e, g, the directions of spontaneous magnetizations for the tetragonal L6 0 nanoprecipitates are distinct from the surrounding BCC matrix phase at the thermally demagnetized state (0 T). Determined from the factor sin 2 α in the form NV 2 p FðqÞ j j 2 Δρ 2 mag sin 2 α, the peak in the I-q x curve arises from the magnetic contrast between the magnetizations of the L6 0⊥ nanoprecipitates and the vertical components of the matrix magnetizations. The peak in the I-q y curve then arises from the magnetic contrast between the magnetizations of the L6 0// nanoprecipitates and the parallel components of the matrix magnetizations. According to the literature 51,52 , the different magnetization directions of the nanoprecipitates and matrix can engender strong magnetic scattering. When the field increases to 0.15 T (Fig. 6h), the magnetizations in the L6 0// nanoprecipitates flip toward H together with the rotation of the ferromagnetic domains of the matrix. This results in magnetic contrast collapse in the I-q y curve, i.e., the disappearance of the peak at 0.15 T. Moreover, the magnetizations of the L6 0⊥ nanoprecipitates remain far from H since the peak still exists in the I-q x curve, indicating that 0.15 T is not sufficiently strong to switch the magnetizations of the L6 0⊥ nanoprecipitates along the field direction, which is consistent with the L-TEM observation in Fig. 5i, where the DWs of the matrix disappear and those of several L6 0 nanoprecipitates still exist. Following the literature 48,49 , the field dependence of the aspect ratio of scattering intensity [I(q y )/I(q x )] summed within the peak range (0.1 nm −1 < q < 0.3 nm −1 ) is plotted in Fig. S8b. Since the magnetizations of the L6 0// and L6 0⊥ nanoprecipitates may simultaneously rotate with the ferromagnetic domains of the matrix, the aspect ratio will increase with increases in the magnetic field owing to the increased magnetic scattering in the direction perpendicular to the field. However, the aspect ratio gradually decreases to~0.36 as the field increases to 0.15 T, which confirms the nonuniform flipping of the magnetizations of the L6 0// and L6 0⊥ nanoprecipitates. The increased magnetic contrast between the magnetizations of the L6 0⊥ nanoprecipitates and the vertical components of the matrix magnetizations gives rise to the relatively enhanced scattering intensity for I-q x , and the reduced magnetic contrast between the magnetizations of the L6 0// nanoprecipitates and the parallel components of the matrix magnetizations gives rise to the relatively decreased scattering intensity for I-q y . When the field further increases to 0.5 T (Fig. 6i), the magnetizations of the L6 0⊥ nanoprecipitates are also nearly oriented along the field direction, as determined from the magnetic contrast collapse in the I-q x curve and the further increase in the aspect ratio (Fig. S8b). Such results indicate that the large magnetostriction generated below 0.15 T arises mainly from the magnetization flips of the ferromagnetic domains of the matrix and the L6 0// nanoprecipitates. Considering that the magnetization flips of the L6 0⊥ nanoprecipitates should result in larger magnetostriction than those of the L6 0// ones, the SANS measurements then exclude the contribution of the magnetization flips of the L6 0⊥ nanoprecipitates to the large magnetostriction obtained in the present Fe-Ga alloys.

Discussion
The present findings may have important implications. First, this work may provide important insights toward understanding the microstructure origin of the large magnetostriction in Fe-Ga alloys. In fact, it is still an open question why soluting nonmagnetic Ga into Fe can induce a significant magnetostriction enhancement. In particular, intrinsic factors, namely, Ga-induced changes in the electronic structures of the A2 solid solution, were proposed to explain this phenomenon [53][54][55] . Note that Fe-Ga alloys are inherently heterogeneous at the nanoscale. Both macroscopic and local techniques have identified that in addition to the A2 matrix 26,[34][35][36][37][38]56,57 , quenched Fe-Ga alloys also contain short-range-ordered Ga-Ga pairs (<2 nm), forming cubic D0 3 nanoprecipitates with Ga-Ga pairs along the <110> axis of the A2 matrix and/or tetragonal L6 0 -type nanoprecipitates with Ga-Ga pairs along the <001> BCC axis. It is clear that the cubic D0 3 nanoprecipitates are detrimental since slow cooling promotes the long-range-ordering of the D0 3 phase and deteriorates magnetostriction [39][40][41][42] . It was shown that proper heat treatment can avoid the formation of D0 3 nanoprecipitates, thereby effectively enhancing the magnetostriction of Fe-Ga alloys. For instance, λ //s of the Fe 81 Ga 19 alloy can be enhanced by 71.4% when quenched near the phase boundary between A2 and (A2 + L1 2 ) after slow cooling (free of D0 3 nanoprecipitates), in comparison with the D0 3 -bearing sample quenched at high temperature (1273 K) 58 . Comparably, the present approach of shortterm aging can provide a more significant magnetostriction enhancement (e.g., magnetostriction increased 3-fold after aging the Fe 81 Ga 19 alloy for 8 h), given that harmful D0 3 ordering occurs simultaneously with the formation of L6 0 nanoprecipitates. This indicates that the L6 0 -type nanoprecipitates should play an important role in enhancing magnetostriction in addition to the intrinsic factors. The present work contributes to finding a feasible approach to enlarge the size and enrich the population of L6 0 -type nanoprecipitates that strongly affect the magnetostriction of Fe-Ga alloys, thus giving rise to substantial magnetostriction enhancement. In addition, our work also helps to understand why L6 0 -type nanoprecipitates form in Fe-Ga alloys. During rapid cooling, the high vacancy density leads to the short-range-ordering of Ga-Ga pairs along the <001> direction, forming very fine L6 0 nanoprecipitates. When subjected to further aging, the fine L6 0 nanoprecipitates grow toward forming the L1 2 equilibrium phase, which requires simultaneous atomic position changes and lattice distortion of the matrix. In the early transformation stage, the L1 2 precursors are constrained at the nanoscale by the untransformed matrix. As the aging time extends, equilibrium L1 2 particles are preferably formed in local regions. Because of its intrinsic negative magnetostriction, it will offset the positive magnetostriction of the distorted matrix, leading to a decrease in net magnetostriction, as shown in Fig. 1. Consequently, the optimum condition to obtain large magnetostriction in Fe-Ga alloys is to form L6 0 nanoprecipitates in the intermediate state. It should also be noted that this diffusion-controlled phase transformation is highly sensitive to the material dimensions since the atomic diffusion rate is much faster at the material surface than at the internal material surface. When utilizing this strategy to obtain large magnetostriction in Fe-Ga alloys, the optimum aging time for a fixed composition depends on the sample size, for instance, the thickness of sheet samples. As shown in Fig.  S9, after aging for 1 h at 723 K, the surface of a 1 mmthick Fe 73 Ga 27 sheet sample already contained mainly the L1 2 phase.
Second, this work provides important insights into the heterogeneous magnetostriction mechanism of ferromagnetic composites with nanoscale precipitation. Early investigations have suggested that the matrix surrounding the nanoprecipitates rather than the nanoprecipitates themselves is the source of the large magnetostriction of Fe-Ga alloys 36,37,49,56 . Within the tetragonal nanoprecipitate model proposed by Khachaturyan et al. 59 , the tetragonal nanoprecipitates rotate by realigning the c-axis or distort in concert with the much larger ferromagnetic domains, and elastic coupling of nanoprecipitates to the matrix results in large magnetostriction. Jiang et al. 36 suggested that the nanoprecipitates already generated tetragonal distortion of the cubic matrix and that rotation of the matrix tetragonality results in large magnetostriction. Our work supports the local tetragonal distortion of the matrix. The distorted matrix shares a general morphology with premartensitic tweeds or unfrozen strain glass [60][61][62] , which contain a very fine mixture of variously oriented spontaneous strain states at the nanoscale. According to the model proposed by Seiner et al. 61 , reorientation of the premartensitic tweeds in Ni-Mn-Ga alloy can be easily induced by a small magnetic field with a magnetostriction of~200 ppm. Similar low-field-triggered large magnetostriction has also recently been reported in ferromagnetic Fe-Pd strain glass with tetragonal strain nanodomains embedded in a cubic matrix 32 . The presently aged Fe-Ga alloys with local tetragonal distortion are new members of this family. In addition, our in situ SANS investigations suggested that the magnetization flips of the L6 0 nanoprecipitates (those with magnetizations perpendicular to the field direction) occur at high fields, thereby excluding their direct role in the large magnetostriction at low fields. A recent phase field simulation work 37 is consistent with this experimental finding, in which the Ga-Ga pairs (along the [001] easy axis) of the L6 0 nanoprecipitates are fixed under magnetic fields.
Third, our work provides a feasible approach to overcome the trade-off between large magnetostriction and the small switching field of magnetostrictive materials.
Although inducing uniaxial anisotropy by applying prestress or by magnetic annealing to redistribute the magnetic domains is effective in approaching the magnetostriction limit of a given material, it inevitably enlarges the switching field because the external field should first overcome the induced extra energy. The present work, which introduces mechanically harder tetragonal nanoprecipitates into a softer cubic matrix, overcomes this trade-off. On the one hand, the local tetragonal distortion of the matrix increases the spontaneous magnetostriction constant λ 001 , i.e., the average tetragonality of the matrix. This, can be determined from a comparison of diffraction spots between the as-quenched and optimally aged samples, as shown at the bottom of Fig. 3d, e. On the other hand, although the locally distorted regions or the tetragonal nanoprecipitates may have stronger local MCA than the ideal cubic matrix, the random magnetic coupling at the nanoscale between them and the matrix results in the collapse of net MCA. According to Cullen et al. 15 , <001> Ga-Ga pairs can produce local uniaxial <001> anisotropy. The local regions with <001> Ga-Ga pairs were suggested to form the L6 0 (i.e., modified-D0 3 ) structure 35,36 . Our atomicresolution STEM-HAADF Z-contrast image has directly shown that the L6 0 nanoprecipitates contain <001> Ga-Ga pairs 47  , and 100 Â Ã ) of the BCC matrix due to simultaneous precipitation in local regions. The competition between the coherent crystalline anisotropy of the matrix and the anisotropy created by the fluctuations in the magnetization due to the local uniaxial <001> anisotropy of the L6 0 nanoprecipitates then results in an instability of the <001> uniformly magnetized state and manifests itself by the vanishing anisotropy. Following this, the locally distorted regions (strain nanodomains) may play a similar role as the L6 0 nanoprecipitates. However, how the magnetic couplings among the strain nanodomains, the L6 0 nanoprecipitates and the undistorted matrix quantitively affect the net anisotropy calls for further study. Since random short-range magnetic coupling is also known to improve magnetic susceptibility 14,63 , the nanodispersive precipitates within the matrix can sensitively respond to low external fields, which accordingly reduces the switching field. With further development, this knowledge can be applied to obtain large sensitive magnetostriction in other nanostructured ferromagnets. We note that in Fe-Co alloys, quenching at the phase boundary between BCC and FCC can also create tetragonal nanoprecipitates, giving rise to larger and more sensitive magnetostriction when compared with the single BCC state 3,64-66 . Consequently, introducing dispersive tetragonal nanoprecipitates into a cubic matrix can be a universal approach to achieve large and sensitive magnetostriction. For engineering applications, heat treatment (short-term aging or quenching at specific temperatures) is a very simple approach when compared with single crystal growth and magnetic annealing approaches, which usually require special instruments and greater energy consumption. In addition, when compared with the approach of applying prestress, the present approach also has advantages in simplifying the design of magnetostriction-associated devices.

Conclusions
Our work provides a feasible approach to obtain large and sensitive magnetostriction in random polycrystalline ferromagnets by aging-induced nanodispersive precipitates. As shown in the proof-of-principle reference, Fe-Ga alloys, short-term isothermal aging of the BCC precursor can effectively introduce FCT L6 0 nanoprecipitates. Manipulating the size and density of the L6 0 nanoprecipitates significantly enhances the magnetostriction of the random polycrystals to be comparable to that for single crystal materials and effectively reduces the net magnetocrystalline anisotropy. The structural investigations reveal that the harder nanoprecipitates create local tetragonal distortion of the softer matrix (i.e., increasing the spontaneous tetragonal magnetostriction constant λ 001 ), resulting in magnetostriction enhancement. In situ magnetic domain structure observations and SANS measurements reveal that random magnetic coupling at the nanoscale effectively improves the magnetic susceptibility, hence giving rise to a 5-fold enhancement in the figure of merit. These findings not only provide important insights toward understanding the microstructural origin of the extraordinary magnetostriction enhancement induced by soluting nonmagnetic Ga into Fe but also help to understand the heterogeneous magnetostriction mechanism. Aging-induced nanodispersive tetragonal precipitates within a cubic matrix can be a feasible approach to achieve large and sensitive magnetostriction in ferromagnetic materials.