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Xiaoqin Ke, Chao Zhou, Ben Tian, [Yoshitaka Matsushita](https://orcid.org/0000-0002-4968-8905), [Xiaobing Ren](https://orcid.org/0000-0002-4973-2486), Sen Yang, Yunzhi Wang

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[Direct evidence of magnetization rotation at the ferromagnetic morphotropic phase boundary in <math>  <mrow>    <msub>      <mi>Tb</mi>      <mrow>        <mn>1</mn>        <mo>−</mo>        <mi>x</mi>      </mrow>    </msub>    <msub>      <mi>Dy</mi>      <mi>x</mi>    </msub>    <msub>      <mi>Fe</mi>      <mn>2</mn>    </msub>  </mrow></math> system](https://mdr.nims.go.jp/datasets/43b79f97-3c6c-4817-ad80-9c52edcd74df)

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Direct Evidence of Magnetization Rotation at the Ferromagnetic Morphotropic Phase Boundary in Tb1-xDyxFe2 SystemXiaoqin Ke, 1 Chao Zhou, 1 Ben Tian, 1 Yoshitaka Matsushita, 2 Xiaobing Ren,3 Sen Yang,1,* Yunzhi Wang41School of Physics, MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China2National Institute for Materials Science, Beamline BL15XU, Spring-8, 1-1-1 Kohto, Sayo-cho, Hyogo 679-5148, Japan3Ferroic Physics Group, National Institute for Materials Science, Tsukuba 305-0047, Ibaraki, Japan4Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USAAbstractThe large magnetostriction at ferromagnetic morphotropic phase boundary (MPB) relies on easy magnetization switching under external magnetic fields. It has been proposed that both domain wall motion and magnetization rotation occur under external magnetic fields at the ferromagnetic MPB. However, direct experimental evidence of the latter is still lacking. Here we report direct evidence of both magnetization rotation and domain wall motion under external magnetic field at the MPB of Tb1-xDyxFe2 system through in-situ synchrotron XRD experiments, which are further confirmed by phase field simulations. This work unravels the origin of the large magnetostriction at ferromagnetic MPB and could shed light on the design of magnetostrictive materials. *corresponding author (yang.sen@mail.xjtu.edu.cn)1. INTRODUCTIONFerromagnetic morphotropic phase boundary (MPB), which was initially named as “spin reorientation boundary” [1], refers to the phase boundary separating tetragonal structure with [001] easy magnetization direction and rhombohedral structure with [111] easy magnetization direction in the phase diagram of ferromagnetic systems [2-3]. Such ferromagnetic MPB has been found in a number of binary ferromagnetic systems such as Tb1-xDyxFe2, Tb1-xDyxCo2, and Tb1-xGdxFe2 [2-5]. Similar to the giant piezoelectricity at ferroelectric MPB [6-7], giant magnetostriction has been reported at ferromagnetic MPB [2-3], which could find applications in a wide range of devices such as sensors, actuators, transducers and sonars and, thus, attracts much attention in recent years. The magnetostriction of ferromagnetic materials is underpinned by magnetic domain switching under an external magnetic field. Therefore, to understand the giant magnetostriction at ferromagnetic MPB, it is essential to know how domain switching occurs under external magnetic fields at MPB. Similar to the physically-parallel ferroelectric MPB systems where two possible domain switching models exist, i.e., polarization domain wall motion [8] and polarization rotation [9], there are also two existing domain switching models at ferromagnetic MPBs, i.e., magnetic domain wall motion and magnetization rotation [1, 3-5,10-18]. While the magnetic domain wall motion mechanism has been theoretically predicted and experimentally confirmed [10-13], there has been no direct experimental evidence to support the magnetization rotation mechanism although it was proposed theoretically over fifty years ago [1,18]. This may greatly hinder the understanding on the origin of large magnetostriction at ferromagnetic MPBs. One possible way to experimentally detect the magnetization rotation under external magnetic fields is to measure the lattice parameter change under external fields, as the crystal lattice structure is directly coupled to the magnetization direction [19]. However, unlike the strong polarization-lattice coupling in ferroelectric systems and thus large lattice distortion at ferroelectric phase transitions [20], ferromagnetic materials generally show very small lattice distortion at ferromagnetic phase transitions due to the weak spin-lattice coupling in ferromagnetic systems [19]. Thus, it has remained as a big challenge to detect magnetization rotation experimentally.In this study, we consider the well-known ferromagnetic MPB system Tb1-xDyxFe2 [3, 17-18, 21] and carry out in-situ high-resolution synchrotron XRD measurements on the ferromagnetic MPB composition at different temperatures with 0 and 4k Oe external magnetic fields. We found that under the external magnetic field, only domain wall motion occurs at temperatures far away from the MPB as evidenced by the intensity change of characteristic XRD peaks without any shift of peak positions. However, at temperatures near the MPB, in addition to domain wall motion, magnetization rotation also occurs, which is supported by the position shift of characteristic XRD peaks. Further phase field simulations have confirmed that both domain wall motion and magnetization rotation occur at the ferromagnetic MPB due to the small magnetization anisotropy. This work unravels the mechanism of domain switching at ferromagnetic MPB and could shed light on the design of new magnetostrictive materials.II METHODSA. Experimental methodsThe Tb1-xDyxFe2 alloys (x=0.0-1.0) were prepared by arc melting from high-purity (99.9%) Tb, Dy, and Fe in argon atmosphere. The crystal structure was observed by high-resolution synchrotron XRD at the BL15XU NIMS beamline in Spring-8. The samples for XRD measurement were first grounded into powders, and then sealed into quartz capillaries with a diameter of 1mm. During the XRD measurements, the capillaries were rotated so that the effect of possible preferred orientation can be reduced and the diffraction density can be averaged. During XRD measurements the temperatures of the samples were controlled to vary between 40 and 400 K by a blow-typed cryocooler and the external magnetic fields were applied through a NdFeB permanent magnet placed under the rotating sample [19]. The magnetic properties of the samples were tested by SQUID, and the magnetostriction of them was measured utilizing strain gauges. The storage modulus of the samples was measured by the DMA equipment at a frequency of 1Hz.B. Phase field simulation methodsPhase field simulations of Tb0.3Dy0.7Fe2 single-crystalline sample were performed. The magnetization vector of each domain in the system is represented by M (M=MS m), in which MS is the saturation magnetization and m (m1, m2, m3) is the unit vector describing the magnetization direction. The total free energy of the system Ftotal is written as the sum of the magnetocrystalline anisotropy energy Fani, the exchange energy Fexch, the magnetostatic energy Fmag, the external magnetic energy Fex, and the elastic energy Fel, i.e., . The magnetocrystalline anisotropy energy density fani is written as a function of (m1, m2, m3):                                   (1)where K1 and K2 are the magnetocrystalline anisotropy coefficients. The exchange energy density fexch is written in terms of m as follows: . The magnetostatic energy density fmag is calculated by , in which the total stray fieldis the sum of the long range spin-spin interaction field and the demagnetization field .  is solved by the magnetostatic equilibrium equation . The external magnetic energy density fex is written as: , where Hex is the external magnetic field. The elastic energy density fel is calculated by , where cijkl is the elastic stiffness tensor, eij is the elastic strain,  is the total strain and is the spontaneous strain or stress-free strain. The stress-free strain can be calculated as:                                            (2)where  and  are the magnetostrictive coefficients of a cubic crystal.The magnetization evolution is then obtained by solving the time dependent Ginzburg-Landau equation: [22], where L is the kinetic coefficient. The parameters used in our simulations are: MS=8.0×105 A/m, K1=-1.2×104(T/K-250) J/m3 (in which T is the temperature), K2=0, λ100=100ppm, λ111=1600ppm, c11=1.41×1011 N/m2, c12=6.48×1010 N/m2, c44=4.87×1010 N/m2 [23]. The simulations were carried out in two dimensions with cell sizes of 512×512 grids. Periodic boundary conditions were applied in both dimensions. The time-dependent Ginzburg-Landau equation was solved by the semi-implicit Fourier spectral method.III RESULTS AND DISCUSSIONSA. Full phase diagram of Tb1-xDyxFe2 system and the magnetostrictive properties at MPBFig. 1(a) shows the temperature-composition phase diagram of Tb1-xDyxFe2 system, which was constructed using magnetization-temperature curves of all compositions combined with the ac susceptibility()-temperature curves and synchrotron XRD profiles of MPB compositions. Note that although extensive work has been done on this ferromagnetic MPB system [3, 17-18, 21], this is the first time that a complete phase diagram has been constructed. The phase diagram consists of four regions: a high-temperature cubic (C) phase (paramagnetic) region, a low-temperature rhombohedral (R, easy magnetization direction <111>) region, a low temperature tetragonal (T, easy magnetization direction <100>) phase regions, and an intermediate state (IS) region in between the R and T phase region. The Curie temperatures (TC) of all compositions in the phase diagram are determined from the magnetization-temperature curves of different compositions as exemplified by that of Tb0.3Dy0.7Fe2 given in Fig. 1(b). The rhombohedral (R) to tetragonal (T) phase transition temperatures (TMPB) of MPB compositions are obtained from the  versus temperature relations as exemplified by that of Tb0.3Dy0.7Fe2 given in Fig. 1(c). The inset figure in Fig. 1(c) also gives the storage modulus versus temperature curve of Tb0.3Dy0.7Fe2, where the storage modulus exhibits a dip at TMPB, indicating elastic softening at the MPB and is similar to the elastic softening found at ferroelectric MPBs [24]. The shaded area at the MPB line corresponds to an intermediate state (IS) between the R and T phase regions detected by the synchrotron XRD results given in Fig. 1(d), which shows that with temperature decreasing, the {440} diffraction pattern evolves from that of R (two {440} peaks) at 300 K to that of T (one {440} peak only due to the too small tetragonal distortion here) at 160 K through that of an intermediate state (three {440} peaks) near TMPB (between 250 K and 220 K). Fig. 1 (a) Full phase diagram of TbxDy1-xFe2 system. (b) M-T curves of one representative MPB compositions Tb0.3Dy0.7Fe2. (c) ac susceptibility () versus T curve of Tb0.3Dy0.7Fe2. The inset figure shows the storage modulus versus T curve of Tb0.3Dy0.7Fe2. (d) The synchrotron XRD results at {440} peaks at different temperatures for Tb0.3Dy0.7Fe2.The three {440} peaks appearing in the IS region of the phase diagram could arise from the presence of either a mixture of R and T phases [3] or a mixture of R (or T) and monoclinic (M) phases. Fig. 2 shows the comparison of the XRD Rietveld refinement results by the model of T+R mixture and the model of R+M mixture for {440} peaks at 250K, which illustrates that the values of goodness-of-fit(chi2) and Rietveld indices(Rwp) are small for both two models. Thus it is difficult to distinguish whether the new M phase exist at MPB through the synchrotron XRD results at zero field only. However, both the phase field simulation results and the synchrotron XRD data under external magnetic fields given below suggest the presence of the new M phase. Also note that for the typical ferroelectric MPB systems such as PbMg1/3Nb2/3O3-xPbTiO3 and PbZrO3-xPbTiO3, both experiments and theoretical models have indicated the presence of the M phase at the MPBs [9, 25]. Fig. 2 Comparison of results of the XRD Rietveld refinement by the model of rhombohedral (R) + tetragonal (T) mixture and the model of rhombohedral(R) + monoclinic (M) mixture. Note that here cubic(C) phase instead of tetragonal (T) phase is used for fitting because the tetragonal distortion in Tb1-xDyxFe2 system is too small to be resolved by synchrotron XRD. Fig. 3(a1)-(b1) shows the measured strain-magnetic field loops at different temperatures and the strain-temperature curves for Tb0.3Dy0.7Fe2 under an external magnetic field of 10 kOe and 50 kOe, respectively. Fig. 1(a2) and (b2) summarizes the variation of its magnetostriction at these two fields with temperature. It indicates that under an external magnetic field of 10 kOe, the largest magnetostriction occurs at ~220 K and reaches a value over 1200 ppm. Under a larger external magnetic field of 50 kOe, the largest magnetostriction occurs at a lower temperature of ~200 K and reaches a value of ~1400 ppm. Therefore, it is clear that the largest magnetostriction appears near the MPB (at the IS region) of the Tb1-xDyxFe2 phase diagram. Fig. 3 The magnetostriction of Tb0.3Dy0.7Fe2 at different temperatures under an external field of 10 kOe ((a1) and (a2)) and 50 kOe ((b1) and (b2)).B. Direct evidences of magnetization rotation at the MPB composition (Tb0.3Dy0.7Fe2) detected by synchrotron XRD experimentsIf magnetization rotation occurs under external magnetic field, lattice distortion would be induced due to the magnetoelastic coupling effect shown by equation (2). It can be deduced from equation (2) that, if the magnetization direction is along [001] direction, , . Such a lattice distortion produces a tetragonal symmetry. When the magnetization direction is along [111] direction, ,, which corresponds to a rhombohedral symmetry. When the magnetization direction is along [110] direction, ,,, which corresponds to an orthorhombic symmetry. Finally, when the magnetization direction is along [uuv] or [0uv] direction, ,, which corresponds to a monoclinic symmetry. Therefore, in order to find direct evidences of magnetization rotation, we then investigate the domain switching mechanism of the Tb0.3Dy0.7Fe2 MPB composition through comparison of synchrotron XRD peaks under zero and a finite magnetic field. Fig. 4 shows the selected regions of in-situ synchrotron XRD patterns obtained both under zero field and under 4k Oe field at 300 K (in the R phase region) and 160 K (in the T phase region), which demonstrate that at 300 K, the intensity ratio for (222)/ (222 (_)) peaks shows a large increase after the external magnetic field is applied. This intensity change is due to the domain reorientation through domain wall motion, which is also reflected in the increased (440)/ (44 (_)0) peak intensity ratio [19]. On the other hand, these XRD patterns also show that the positions of all three characteristic peaks do not shift, which suggests that the lattice parameter of the crystal lattice does not change. Similarly, at 160 K no peak shift occurs before and after the external magnetic field is applied, thus also suggesting no lattice parameter change. Therefore, at temperatures far away from MPB, both in the R phase and T phase regions, the external magnetic field reorient the domains through domain wall motion but does not change the lattice parameter. Fig. 4. {222},{440},{800} peak profiles of Tb0.3Dy0.7Fe2 at H=0 Oe and H=4 kOe at temperatures in the R phase region (T=300 K) and T phase region (T=160 K). At the T phase region (T=160 K), theoretically both {440} and {800} peaks was supposed to split into two peaks with an intensity ratio of 1:2 and 2:1, respectively. But here the peak splitting does not occur because the tetragonal distortion in the system is too small to be detected. However, at temperatures close to the MPB, the situation is different. Fig. 5 illustrates the selected regions of the XRD patterns of Tb0.3Dy0.7Fe2 both under zero magnetic field and under a magnetic field of 4 kOe at 250K, 230K, 215K and 200K, respectively. As illustrated above in Fig. 1, at these temperatures the structure of the sample at zero external magnetic field is in the intermediate state (IS) region. Under the external magnetic field of 4 kOe, it is demonstrated that at all four temperatures the intensity ratio of {222} and {440} peaks changes significantly, which thus indicates that domain wall motion occurs at all four temperatures. Furthermore, shifts of peak positions have also been detected, which indicates that lattice distortion and thus magnetization rotation through monoclinic (M) phase occur at all four temperatures. Note that at 250 K and 230 K, only the positions of {222} peaks shift for an appreciable amount while at 215 K and 200 K, both {222} peak and {440} peak shift. Such difference could be due to the fact that at higher temperatures (250 K and 230 K) that are closer to the R phase region, the magnetization vector rotates through MB phase with [v u u] (v < u) easy magnetization direction within the (0 1 1) plane while at lower temperatures (215 K and 200 K) that are closer to the T phase region, the magnetization vector rotates through MC phase with [0 u v] easy direction within the (0 0 1) plane [27-28]. Also note that no peak shifts are detected for {800} peaks at all four temperatures because λ100 is extremely small in this system [3]. Therefore, the synchrotron XRD results given in Fig. 5 demonstrates that near TMPB of Tb0.3Dy0.7Fe2 alloy, both magnetization rotation and domain wall motion occur under external magnetic fields. Fig. 5 {222}, {440} and {800} reflections for Tb0.3Dy0.7Fe2 alloy at zero external field and under a field of 4k Oe at temperatures in the intermediate region (at 250K, 230K, 215K, and 200K).C. Confirmation of magnetization rotation mechanism at ferromagnetic MPB by phase field simulationsIn order to illustrate the details of the above two magnetization switching processes, 2D phase field simulations are performed. Fig. 6(a1)-(a4) show the simulated domain structures in Tb0.3Dy0.7Fe2 at different temperatures, represented by the magnetization direction θ of each domain at each grid point, where θ is defined as the angle between the magnetization vector m and the horizontal direction. Fig. 6 (b1)-(b4) illustrate the distribution of T, R and M phases within the whole sample at different temperatures. These simulation results clearly illustrate that upon cooling the domain structure of the sample changes from one with R domains (a1, b1) to a mixture of M+R domains (a2, b2) to a mixture of M+T domains (a3, b3) and finally to one with T domains (a4, b4), which suggests that the IS region detected by synchrotron XRD patterns should include the new M phase. Note that the M phase is not expected according to the magnetocrystalline anisotropy energy because the magnetocrystalline energy used in the simulations takes the form of 6th order polynomial with K1 and K2 only (See equation (1)) [26-28]. However, the M phase still appears because it is stabilized by the long-range elastic energy and magnetostatic energy considered in the simulations under small magnetocrystalline anisotropy, which is similar to our previous finding that the M phase, although not stabilized by the 6th order Landau free energy, could be stabilized by the long-range electrostatic and elastic energy under small polarization anisotropy at ferroelectric MPB [29-30]. Fig. 6. Evolution of domain structure and phases with temperature decreasing for Tb0.3Dy0.7Fe2 alloy obtained by phase field simulations. (a1)-(a4) Domain structure. Different colors represent the angle between magnetization vector and horizontal axis (θ) as indicated by the color bar. (b1) - (b4) The distribution of T, M, and R phases in the sample. Fig. 7(a1)-(c2) then give the domain structure evolution upon the application of an external magnetic field of ~4 kOe at three different temperatures (T=315K, T=255K, T=195K) and Fig. 7(a3)-(c3) plots variation of the magnetization direction (θ) with position along the dashed lines in Fig. 7(a1)-(c2). It is readily seen that with the application of the external magnetic field, only domain wall motion occurs at T=315 K and 195 K (which are away from MPB) while both domain wall motion and magnetization rotation occur at T=255 K (which are near MPB). Such phase field simulation results are qualitatively consistent with the synchrotron XRD results shown in Fig. 4 and Fig. 5. Fig. 7 (a1)-(c1) The domain structure at H=0. (a2)-(c2) The domain structure at H=4 kOe.(a3)-(c3) Variation of θ with distance along the dashed line in (a1)-(c1) and (a2)-(c2). (d1)-(d2) Variation of magnetization direction (θ) under the same external magnetic field at large anisotropy(K1=K0) and small anisotropy(K1=0.1K0), respectively. MR and DWM are the abbreviation of magnetization rotation and domain wall motion, respectively. The dark golden and dark green solid rectangles in (a1)-(c1) and (a2)-(c2) give the positions where DMW and MR represented by the arrows in (a3)-(c3) occurs, respectively.D. Origin of magnetization rotation mechanism and its implicationsTo further understand why continuous magnetization rotation could occur at ferromagnetic MPB, a simple analysis is given below. The total free energy density of the system(ftotal) can be approximated as the sum of magnetocrystalline anisotropy energy and the Zeeman energy, i.e., where  (2D case), and it can be shown that when K1 is large (K1=K0>0), the stable phase is always T phase () both without and with an external magnetic field H1, as shown in Fig. 8(a), but when K1 is small (K1 =0.1K0), the external magnetic field of H1 would rotate the magnetization from the easy [0 1] direction (, T phase) to [u v] direction (, M phase) as shown in Fig. 8(b). Therefore, at ferromagnetic MPB where the magnetocrystalline anisotropic energy coefficient is small, magnetization rotation under external magnetic fields could occur.Fig. 8 Variation of magnetization direction (θ) under the same external magnetic field at large anisotropy (K1=K0) (a) and small anisotropy (K1=0.1K0) (b), respectively.Note that at a ferroelectric MPB, continuous polarization rotation also occurs due to the small polarization anisotropy [9, 30]. It thus can be concluded that continuous rotation of the order parameter vector could be a general domain switching mechanism in ferroic materials with small anisotropy. The conventional domain switching mechanism, i.e., domain wall motion, occurs at ferroic materials with relatively large anisotropy. At small anisotropy, both domain wall motion and continuous rotation of order parameters could be the possible domain switching mechanisms under external fields.IV CONCLUSIONSIn summary, we have constructed a complete phase diagram of TbxDy1-xFe2 system. At the MPB in between the high-temperature rhombohedral phase region and the low-temperature tetragonal phase region, distinct structure and superior magnetostrictive properties are observed. Through synchrotron XRD measurements both under zero and finite magnetic fields, we find that near the MPB, both domain wall motion and magnetization rotation occur under an external magnetic field, contributing to the large magnetostriction. Further phase field simulations confirm that a monoclinic phase appears at the MPB region and magnetization rotation could occur at MPB due to the small magnetization anisotropy. This work suggests that rotation of the vectorial or tensorial order parameters (polarization, magnetization and strain) could be a universal phenomenon for ferroic materials with small crystalline anisotropy that is responsible for the appearance of giant response at MPBs. It could shed light on future design of high-performance ferroic materials. AcknowledgementX.K. and S.Y. acknowledge the supports by the National Key R&D Program of China (2021YFB3501401 and 2022YFE0109500), the National Natural Science and Foundation of China (Grant Numbers 52171189 and 12374120), Key Scientific and Technological Innovation Team of Shaanxi province (Grant Number 2020TD-001), Innovation Capability Support Program of Shaanxi (Grant Numbers 2018PT-28, 2017KTPT-04), Shanghai Aerospace Science and Technology Innovation Fund (SAST2018-117), the Fundamental Research Funds for the Central Universities (China), the World-Class Universities (Disciplines), the Characteristic Development Guidance Funds for the Central Universities. YW acknowledge the support by the US Natural Science Foundation under Grant No. DMR-1923929, which has facilitated this international collaboration.References[1] H. Horner and C. M. Varma, Nature of spin-reorientation transitions, Phys. Rev. Lett. 20,845(1968).[2] S. Yang, H. Bao, C. Zhou, Y. Wang, X. Ren, Y. Matsushita, Y. 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