# Fileset

[J. Appl.Phys.136,033904(2024).pdf](https://mdr.nims.go.jp/filesets/e8512675-e890-4e6d-a4aa-7e246f7fa23d/download)

## Creator

[Masamichi Nishino](https://orcid.org/0000-0002-2060-2303), [Hiroshi Hayasaka](https://orcid.org/0000-0001-5847-9948), [Seiji Miyashita](https://orcid.org/0000-0003-0681-3910)

## Rights

[Creative Commons BY-NC-ND Attribution-NonCommercial-NoDerivs 4.0 International](https://creativecommons.org/licenses/by-nc-nd/4.0/)

## Other metadata

[Thermodynamic properties of R2Fe14B (R=Dy, Nd) and dysprosium random substitution effect on coercivity in neodymium permanent magnets](https://mdr.nims.go.jp/datasets/cd675c30-a514-4ff3-80c3-ca5318e4c3b0)

## Fulltext

Thermodynamic properties of R2Fe14B (R=Dy, Nd) and dysprosium random substitution effect on coercivity in neodymium permanent magnetsViewOnlineExportCitationRESEARCH ARTICLE |  JULY 18 2024Thermodynamic properties of R2Fe14B (R=Dy, Nd) anddysprosium random substitution effect on coercivity inneodymium permanent magnetsMasamichi Nishino   ; Hiroshi Hayasaka  ; Seiji Miyashita J. Appl. Phys. 136, 033904 (2024)https://doi.org/10.1063/5.0217917 19 July 2024 00:02:16https://pubs.aip.org/aip/jap/article/136/3/033904/3303758/Thermodynamic-properties-of-R2Fe14B-R-Dy-Nd-andhttps://pubs.aip.org/aip/jap/article/136/3/033904/3303758/Thermodynamic-properties-of-R2Fe14B-R-Dy-Nd-and?pdfCoverIconEvent=citejavascript:;https://orcid.org/0000-0002-2060-2303javascript:;https://orcid.org/0000-0001-5847-9948javascript:;https://orcid.org/0000-0003-0681-3910https://crossmark.crossref.org/dialog/?doi=10.1063/5.0217917&domain=pdf&date_stamp=2024-07-18https://doi.org/10.1063/5.0217917https://servedbyadbutler.com/redirect.spark?MID=176720&plid=2505179&setID=592934&channelID=0&CID=907040&banID=522054965&PID=0&textadID=0&tc=1&rnd=7548385162&scheduleID=2423383&adSize=1640x440&data_keys=%7B%22%22%3A%22%22%7D&matches=%5B%22inurl%3A%5C%2Fjap%22%5D&mt=1721347336634435&spr=1&referrer=http%3A%2F%2Fpubs.aip.org%2Faip%2Fjap%2Farticle-pdf%2Fdoi%2F10.1063%2F5.0217917%2F20057892%2F033904_1_5.0217917.pdf&hc=2947e656e54b67605ccf9eb8e3ef3c66da962b04&location=Thermodynamic properties of R2Fe14B (R=Dy, Nd)and dysprosium random substitution effect oncoercivity in neodymium permanent magnetsCite as: J. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917View Online Export Citation CrossMarkSubmitted: 7 May 2024 · Accepted: 25 June 2024 ·Published Online: 18 July 2024Masamichi Nishino,1,a) Hiroshi Hayasaka,1,2 and Seiji Miyashita3,4AFFILIATIONS1Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044,Japan2Global Research and Development Center for Business by Quantum-AI technology, National Institute of Advanced IndustrialScience and Technology, 1-1-1, Umezono, Tsukuba, Ibaraki 305-8568, Japan3JSR-UTokyo Collaboration Hub, CURIE, Department of Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan4The Physical Society of Japan, 2-31-22 Yushima, Tokyo 113-0033, Japana)Author to whom correspondence should be addressed: nishino.masamichi@nims.go.jpABSTRACTNeodymium (Nd) magnets (Nd2Fe14B) are key materials for achieving high energy conversion efficiency. The coercive forces (fields) of themagnets are often reinforced by adding dysprosium (Dy), especially at high temperatures. To understand the magnetic properties of Dy-sub-stituted systems (Nd1�xDyx)2Fe14B, it is important to study those of Dy2Fe14B and Nd2Fe14B and analyze the difference in detail from amicroscopic viewpoint. Applying a recently developed atomistic model approach, we investigated thermodynamic properties of thesemagnets. We studied the temperature and field dependences of the magnetizations, and anisotropy fields and energies. We found that thesimulation results captured the characteristic features of the experimentally observed data. We discuss the detail with the magnetization pro-files of the component atoms. Furthermore, we investigated the effect of Dy random substitution on the coercivity in two systems: one incontact with vacuum and the other in contact with a grain boundary phase. We found that the threshold fields increased almost linearlywith the concentration of Dy atoms in both systems, which was compared to the results of the layer-by-layer substitution effect analyzed inour previous work. We discuss the influence of the arrangement of Dy atoms on coercivity enhancement.© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/). https://doi.org/10.1063/5.0217917I. INTRODUCTIONControl of the coercivity of permanent magnets and realiza-tion of higher coercivity at higher temperatures is an importantissue for achieving high energy conversion efficiency. Coercivity isa nonequilibrium property that depends on various factors such asgrain boundary properties.1–9 Therefore, understanding the mecha-nism of coercivity remains challenging.Neodymium (Nd) magnets (Nd2Fe14B10–19), known as power-ful permanent magnets with high coercivity, are key materials forachieving high energy conversion efficiency. They are used inmotors, generators, electrical appliances, and other applications.20The use of Nd magnets is expected to increase in the future, partlybecause of the growing demand for electric vehicle (EV) motors.Studies on the coercivity mechanism of Nd magnets and attemptsto achieve higher coercivity at higher temperatures have becomeincreasingly significant.Neodymium magnets have a problem with coercivity at hightemperatures, and the coercivity of the magnets is often reinforcedby adding dysprosium (Dy). The formation of Dy-rich shells,(Nd1�xDyx)2Fe14B, is considered important for the reinforcementof the coercivity. Experiments using the grain boundary diffusionmethod have shown that the coercivity increases without losingremanence,21–25 and the formation of a Dy-rich shell between thegrain boundary and core grain has been observed.25 The coercivityJournal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917 136, 033904-1© Author(s) 2024 19 July 2024 00:02:16https://doi.org/10.1063/5.0217917https://doi.org/10.1063/5.0217917https://pubs.aip.org/action/showCitFormats?type=show&doi=10.1063/5.0217917http://crossmark.crossref.org/dialog/?doi=10.1063/5.0217917&domain=pdf&date_stamp=2024-07-18https://orcid.org/0000-0002-2060-2303https://orcid.org/0000-0001-5847-9948https://orcid.org/0000-0003-0681-3910mailto:nishino.masamichi@nims.go.jphttps://creativecommons.org/licenses/by-nc-nd/4.0/https://creativecommons.org/licenses/by-nc-nd/4.0/https://doi.org/10.1063/5.0217917https://pubs.aip.org/aip/japenhancement effect of the Dy-rich shell was also pointed out incoarse-grained micromagnetics model studies with macroscopicparameters of the Dy-substituted phase.26,27Micromagnetics continuum modeling of permanent magnetshas been applied to the analysis of magnetic properties.28 Thismethod has the advantage of being able to treat large systems andhas had some success in analyzing qualitative aspects of magnetiza-tion reversals. However, because of the coarse graining and applica-tion of macroscopic magnetic parameters, the microscopic detailsof crystal structures and magnetic parameters are ignored. In addi-tion, as investigated in Ref. 29, it is difficult, in principle, to treataccurately temperature effects and thermal fluctuations in coarse-grained continuum modeling.The temperature effects of R2Fe14B have been studied by themean-field approximation,30,31 but the mean-field approximationneglects thermal fluctuations. Thus, it is not sufficient to studyaccurate thermodynamic properties and often leads to serious mis-interpretation. For example, in the S ¼ 1 simple Heisenberg modelwith the magnetic interaction J on a simple cubic lattice (sixnearest neighbors), the mean-field theory leads to the critical tem-perature (2J) around 40% higher than the exact value (1:44J)obtained by treating the fluctuations.32To study the microscopic details of magnetic properties atfinite temperatures, recently developed atomistic models are indis-pensable. In atomistic modeling, microscopic magnetic parameters,which reflect the lattice structure (Fig. 1), thermal fluctuations, anddynamics to realize the thermal equilibrium33,34 can be treatedappropriately. Atomistic model studies have elucidated the qualita-tive and quantitative properties of Nd magnets at zero and finitetemperatures.7,35–52Using an atomistic model approach, we recently studied thecoercivity enhancement by Dy substitution into Nd magnets.50 Wefound that the crystal electric field (CEF) energy barrier of Dyatoms was more resistant to temperature increase, which is consid-ered to be an origin of the coercivity enhancement by Dy substitu-tion, in addition to the difference in the magnetic interactionbetween rare-earth and iron atoms, that is, antiferromagneticcoupling between Dy and Fe moments while ferromagnetic cou-pling between Nd and Fe moments. We also found that an increasein the number of Dy-substituted layers (100% replacement of Ndby Dy) enhanced the coercivity, and the coercivity increase wasnearly proportional to the number of substituted layers.It is important to study Dy2Fe14B and Nd2Fe14B in detail andmake clear the difference in magnetic properties by the atomisticmodel approaches toward a full understanding of the coercivityenhancement mechanism. So far, Nd2Fe14B has been studied usingatomistic models,7,35–50,52 but Dy2Fe14B has not been studied yet.In the present paper, we investigated for the first time the thermo-dynamic properties of Dy2Fe14B using an atomistic modelapproach and compared them with those of Nd2Fe14B. We found asatisfactory agreement with experimental observations. We dis-cussed the detailed features of these magnets.Second, we studied the Dy substitution effect. In our previousstudy (Ref. 50), we investigated the effect of Dy layer-by-layer sub-stitution on the coercivity. In the present paper, we analyzed theeffect of Dy random substitution on magnetization reversal in neo-dymium permanent magnets and compared the results with thoseof the previous study.The rest of the paper is organized as follows. In Sec. II, theatomistic model is presented. In Sec. III, the methods for investigat-ing static and dynamical properties are explained. In Sec. IV A,thermodynamic properties of R2Fe14B (R=Dy, Nd) are studied.In Sec. IV B, the effect of Dy random substitution on magnetiza-tion reversal in neodymium permanent magnets is investigated.Section V is devoted to summary.II. MODELThe following atomistic Hamiltonian was adopted:H ¼ �Xi,j2Jijsi � sj �XFeiDi(szi )2þXRiXl,mΘl,iAml,ihrliiÔml,i � hxXiSxi � hzXiSzi : (1)The first term denotes the exchange interaction Jij between the ithand jth atoms (spins). The second term refers to the anisotropyenergy of Fe atoms, and Di is the anisotropy constant for the ith Featom. The third term represents the crystal electric field (CEF)energy of rare-earth atoms (Nd and/or Dy). The origin of this termis the electrostatic interaction between f electrons of rare-earthatoms and surrounding ions. This term plays an important role inmagnetic properties of rare-earth magnets.53 Here, Θl,i, Aml,i , hrlii,and Ôml,i are the Stevens factor, coefficient of the spherical harmon-ics of the crystalline electric field, an average of rl over the radialwave function, and Stevens operator, respectively. In the fourth andfifth terms, hx and hz are the external fields applied in the x and zdirections parallel to the a and c axes, respectively. We considerl ¼ 2, 4, 6 and m ¼ 0 (diagonal operators), which have the domi-nant contribution.For Fe and B atoms, si denotes the magnetic moment atthe ith site, while for Nd and Dy atoms, it is the moment of thevalence (5d and 6s) electrons and is strongly coupled to theFIG. 1. (a) Side view and (b) top view of the unit cell of R2Fe14B. R denotes arare-earth atom (Nd or Dy). In Nd2Fe14B, la ¼ lb ¼ 8:80 Å, and lc ¼ 12:20 Å.In Dy2Fe14B, la ¼ lb ¼ 8:76 Å, and lc ¼ 12:01 Å.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917 136, 033904-2© Author(s) 2024 19 July 2024 00:02:16https://pubs.aip.org/aip/japmoment of the 4f electrons, J i ¼ gTJ iμB, where gT is the Landég-factor and J i is the total angular momentum. Thus, the totalmoment of a rare-earth atom is Si ¼ si þJ i. For Nd atoms,J ¼ L� S ¼ 9=2 and gT ¼ 8=11, where L is the orbitalangular momentum and S is the spin angular momentum, whilefor Dy atoms, J ¼ Lþ S ¼ 15=2 and gT ¼ 4=3. For the Fe and Batoms, we define Si ¼ si. As shown in Ref. 50, si of a Nd (Dy) atomand Si( ¼ si) of an Fe atom are coupled antiferromagnetically, butSi of a Nd atom and that of an Fe atom are coupled ferromagneti-cally. In contrast, Si of a Dy atom and that of an Fe atom arecoupled antiferromagnetically. Following the previous paper,50 weused the exchange interactions and magnetic moments estimatedusing the Korringa–Kohn–Rostoker (KKR) first-principles method,anisotropy constants for Fe atoms (six types) provided by Ref. 54,and Aml provided by Yamada et al.55 for Nd and Dy atoms inR2Fe14B with hrli estimated by Ref. 56.The lattice constants are close between Nd2Fe14B andDy2Fe14B, and as we reported in Ref. 50, the exchange interactionsestimated by the KKR first-principles computation between Nd andFe in Nd2Fe14B and those between Dy and Fe in Dy2Fe14B werevery close. This suggests that the effect of the local distortions pro-duced by Dy substitution is small, and we do not consider thiseffect in Dy substitution.III. METHODA. Thermodynamical propertiesWe used a Metropolis importance-sampling Monte Carlo(MC) method to study equilibrium magnetizations at finite temper-atures. The per-site magnetizations Mz , Mx , M, and Mxy for theR2Fe14B model were defined asMz(R2Fe14B) ¼ 1NXNi¼1Szi����������* +, (2)Mx(R2Fe14B) ¼ 1NXNi¼1Sxi����������* +, (3)M(R2Fe14B) (4)¼ 1NffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNi¼1Sxi !2þXNi¼1Syi !2þXNi¼1Szi !2vuut* +, (5)andMxy(R2Fe14B) ¼ 1NffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNi¼1Sxi !2þXNi¼1Syi !2vuut* +, (6)respectively. Here, N is the number of all atoms (all spins) in theR2Fe14B model and hi denotes thermal average.We also define the atom-specified (per-site) magnetizationsfor each atom species: R and Fe, (We computed the magnetizationof boron (B) atoms, but the contribution was negligible.)mz(A) ¼ 1NAXNAi¼1Szi����������* +� sgnXNAi¼1Szi !, (7)mx(A) ¼ 1NAXNAi¼1Sxi����������* +� sgnXNAi¼1Sxi !, (8)mxy(A) ¼ 1NAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNAi¼1Sxi !2þXNAi¼1Syi !2vuut* +, (9)m(A) ¼ 1NAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNAi¼1Sxi !2þXNAi¼1Syi !2þXNAi¼1Szi !2vuut* +, (10)where A denotes R or Fe and NA is the number of atom A in theR2Fe14B model.We performed 200 000 Monte Carlo steps (MCSs) for equili-bration and the following 400 000 MCSs for measurement for asystem of 6� 6� 6 unit cells with periodic boundary conditions.B. Dynamical propertiesTo study the threshold values of the field for magnetizationreversal in Dy-substituted Nd magnet systems, we applied the sLLGequation33,34ddtSi ¼ � γ1þ α2iSi � heffi � αiγ(1þ α2i )SiSi �hSi � heffii: (11)Here, γ is the electron gyromagnetic ratio and αi is the dampingfactor.At finite temperatures, magnetization reversal is a barrier-crossing process by virtue of thermal fluctuations, which occurs ina stochastic process. To treat the thermal effect, a noise field wasintroduced into the effective field on the ith asheffi ¼ � @H@Siþ ξiðtÞ: (12)Here, ξi(t) ¼ (ξxi , ξyi , ξzi ) is the white Gaussian noise field, whichhas the following properties:hξμi (t)i ¼ 0, hξμi (t)ξνj (s)i ¼ 2Diδijδμνδ(t � s): (13)Temperature T is described by the following relation (fluctuation–dissipation relation):Di ¼ αikBTγSi: (14)We applied a kind of middle-point method34 equivalent to theHeun method33 for the numerical integration of the equation inJournal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917 136, 033904-3© Author(s) 2024 19 July 2024 00:02:16https://pubs.aip.org/aip/japthe Stratonovich interpretation. We set the time step to Δt ¼ 0:1 fsand αi ¼ 0:1.49Under a reversed field parallel to the z (c) direction (hz = 0and hx ¼ 0), we observed the time evolution of the magnetization~Mz ¼XiSzi (15)starting from an all-down-spin state. According to the previousstudy,49 the threshold magnetic field was determined as follows.For a given value of hz , we performed twelve simulations using dif-ferent random number sequences to generate the noise field. Wecounted the number (Ns) of samples (simulations) in which mag-netization reversal occurred. The threshold field was then definedas the middle point of the interval between the upper limit of thefield for Ns ¼ 0 and the lower limit of the field for Ns ¼ 12. Theerror bars in the threshold field represent the interval region of thefield. We set tmax ¼ 0:5 ns (5� 106 time steps).In experiments, a coercive field is often defined as a field inwhich the metastable magnetic state has a lifetime of 1s, i.e., relaxa-tion time of 1s. However, this is not practical for real-time simula-tions. Because the reversal time increases exponentially around thethreshold field, the threshold fields estimated in this study provideapproximate coercive fields.42IV. RESULTSA. Differences in thermodynamic properties betweenR2Fe14B magnets (R=Nd, Dy)In Fig. 2, the temperature dependences of Mz , M, and Mxyunder zero field for Dy2Fe14B are compared with those ofNd2Fe14B. We found that the critical temperature Tc of Dy2Fe14Bwas almost the same as that of Nd2Fe14B (Tc ≃ 870K). This is pri-marily because the values of Jij were similar for the two magnets asshown in Ref. 50. The simulated critical temperatures (Tc) of themagnets were a little overestimated compared to the experimentalvalues (≃ 600K).13,15,16 We use the temperature in the scaled formT=Tc for comparison with experimental data.The temperature dependences of the magnetizations inDy2Fe14B captured the characteristic features of the experimentallyobserved ones.15,16 When the temperature was reduced,Mz(Dy2Fe14B) increased from Tc and reached the maximum valueof Mz ≃ 0:87 μB at T ≃ 0:58Tc (experimentally, Mz ≃ 0:85 μB atT ≃ 0:66Tc) and decreased gradually to 0 K. Mz ≃ 0:59 μB per siteat 0 K was an approximation of the experimental estimateMz ≃ 0:66 μB at 4.2 K.16Nd2Fe14B exhibited a spin-reorientation transition atTR ≃ 150K in experiments.16–18,55 It was simulated accurately,yielding 2:0 μB/site at TR ≃ 150K, which was close to an experimentalvalue of, 2:1 μB/site at T ¼ TR.16 Dy2Fe14B did not exhibit spin-reorientation transition and a smooth curve in Mz , owing to theanisotropy energy parallel to the c-direction at all temperatures.50We examined the detailed features of magnetizations in eachcomponent of the magnets. In Fig. 3, the temperature dependencesof mz(Dy) and mxy(Dy) under zero field for Dy2Fe14B are com-pared to those of mz(Nd) and mxy(Nd) for Nd2Fe14B. In Fig. 4, thetemperature dependences of mz(Fe) and mxy(Fe) in Dy2Fe14B arecompared to those in Nd2Fe14B. The total magnetic moments ofNd and Fe atoms were ferromagnetically coupled but those of Dyand Fe atoms were antiferromagnetically coupled (Fig. 1 inRef. 50). Thus, mz(Dy) increased in the negative direction with alarge magnitude blow Tc. As the temperature was lowered, themz(Dy) curve exhibited a pattern of first concaving downward,then upwards, and finally downward again, culminating atmz ≃ �10:4, μB at 0 K. However, mz(Nd) increased in the positivedirection, and the growth decreased below the spin-reorientationtransition (T ≃ 150K) and reached mz ≃ 2:3 μB at 0 K. mz(Fe) andmxy(Fe) were almost the same for the two magnets, exceptT , TR ≃ 150K. mz(Fe) and mxy(Fe) in Nd2Fe14B exhibited alarge reduction and increment, respectively, below T ≃ 150K, andthis dependence was reflected in total magnetizations Mz and Mxyin Nd2Fe14B. The contributions of the increasing function of T ,FIG. 2. Temperature dependences of Mz , M, and Mxy at the zero field forDy2Fe14B and Nd2Fe14B.FIG. 3. Temperature dependences of mz(Dy) and mxy (Dy) in Dy2Fe14B, andmz(Nd) and mxy (Nd) in Nd2Fe14B at the zero field.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917 136, 033904-4© Author(s) 2024 19 July 2024 00:02:16https://pubs.aip.org/aip/japthat is, mz(Dy), and the decreasing function of T , that is, mz(Fe),exhibited Mz in Dy2Fe14B with a peak around T ¼ 0:58Tc.Subsequently, we studied the field dependences of the magne-tizations in Dy2Fe14B and Nd2Fe14B. We focused on the magnetiza-tions at the temperature T ¼ 0:46Tc, close to room temperature. InFigs. 5(a) and 5(b), we depict hx and hz dependences of Mx , Mz ,and M in Dy2Fe14B and Nd2Fe14B, respectively. In Dy2Fe14B, Mxincreased linearly up to hx ≃ 18:0 T. Around hx ≃ 18:0 T, the rateof increase of Mx changed and became slowly for hx . 18:0T,while Mz decreased showing a upward convex curve tohx ≃ 21:0T.In the experiments,15,16 Mz vs hz and Mx vs hx were observedless than 1.5 T (up to around 1.0 T), and the field of a crossingpoint of extensions of these lines was determined as the anisotropyfield of Dy2Fe14B, which was hA ≃ 15:0T at room temperature(300 K). There Mz was almost saturated for hz , 1:5 T andconsidered to be a saturated magnetization, i.e., Mz ¼ Msat¼ 0:824 μB/site. In the simulation, Mz gradually increases for hz[Fig. 5(a)]. However, we adopted the same procedure as the experi-ments to estimate hA in the simulation. The saturated Mz wasdetermined as Mz ¼ 0:85 μB=site at hz ¼ 1:0 T, and hA was esti-mated as the field of the crossing point of the line of Mx vs hx andline of Mz ¼ 0:85 μB=site. Then we found hA ≃ 15:0 T which cor-responds to the experimental estimation. We give a discussionabout an alternative estimation later.On the other hand, hA is clearly determined in Nd2Fe14B inthe same way as the experiment.15,16 In Fig. 5(b), Mz was almostconstant and Mx increased linearly up to hx ≃ 6:0T, above whichMx remained constant (fully saturated). This is in agreement withexperimentally estimated values, hA ≃ 6:7 T15,16 at room tempera-ture (300 K).It was observed that the simulation approximately reproducedthe field dependencies of the magnetizations in both Dy2Fe14B andNd2Fe14B, and that Dy2Fe14B had a much larger anisotropy fieldthan Nd2Fe14B.In this study, we estimated the anisotropy energy K1 using thefollowing relation:K1 ¼ 12hAMsat, (16)where Msat is the saturation magnetization. In the experiments,16Msat ¼ 0:824 μB/site and hA ≃ 15:0 T at room temperature (300 K)yielded K1 ≃ 4:22MJ=m3. In the simulation, we used Msat ¼ Mz athz ¼ 1:0 T, i.e., Msat ¼ Mz ¼ 0:85 μB/site, and thus we obtainedK1 ≃ 4:35MJ=m3 with hA ¼ 15:0T. We found that the obtainedK1 corresponds to the experimental K1. However, considering thatMx as a function of hx and Mz as a function of hz graduallyincrease at larger fields, it is difficult to estimate K1 precisely. Wediscuss this point later.In contrast, in Nd2Fe14B, Mx increased linearly up tohx ≃ 6:0 T, above which Mx became saturated. We consideredFIG. 4. Temperature dependences of mz(Fe) and mxy (Fe) in Dy2Fe14B, andmz(Fe) and mxy (Fe) in Nd2Fe14B at the zero field.FIG. 5. hx and hz dependences of Mx , Mz , and M in (a) Dy2Fe14B and (b)Nd2Fe14B. T ¼ 0:46Tc.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917 136, 033904-5© Author(s) 2024 19 July 2024 00:02:16https://pubs.aip.org/aip/japMz ¼ 1:65 μB=site at hz ¼ 1:0T as Msat. Using hA ¼ 6:0T, weobtained K1 ¼ 3:31MJ=m3. In the experiments, Msat ¼ 1:91 μB=siteand hA ¼ 6:7T were estimated at room temperature (300 K),16which yielded K1 ≃ 4:27MJ=m3. The estimated hA, Msat, and hA aresummarized in Table I.To investigate the field dependences of the magnetizations ofthe constituent atoms, we give in Figs. 6(a) and 6(b), hx depen-dences of mz(Dy), mx(Dy), and m(Dy) for low and high fields,respectively. In Fig. 6(c), we show hx dependences of mz(Fe),mx(Fe), and m(Fe) on Dy2Fe14B.We also present in Figs. 7(a) and 7(b), hx dependences ofmz(Nd), mx(Nd), and m(Nd) and hx dependences of mz(Fe),mx(Fe), and m(Fe) on Nd2Fe14B, respectively. Magnetizationmx(Dy) increased in the negative direction up to hx ¼ 18T at aconstant rate and then reduced above hx ¼ 18T at a slower rate.Magnetization mx(Dy) became zero around hx ¼ 98 T andincreased linearly in the positive direction at larger fields. It isworth noting that m(Dy) decreased the upward convexity tohx ¼ 18 T, then linearly to hx ¼ 98T, reached m(Dy)≃ 0 athx ¼ 98 T, and increased linearly above hx ¼ 98T. This indicatesthat a paramagnetic-like state of the Dy moments was realizedaround hx ¼ 98T. With respect to the Fe moment, mx(Fe)increased linearly and mz(Fe) decreased with an upward convexcurve up to 18 T, above which mx(Fe) became saturated. On theother hand, mx(Nd) and mx(Fe) in Nd2Fe14B increased linearly upto hx ¼ 6:0 T and became saturated around Mx ≃ M ≃ 1:25 μB/siteand Mx ≃ M ≃ 1:83 μB/site, respectively. From zero to 6:0T,m(Nd) was slightly reduced, whereas m(Fe) remained constant. InFigs. 8(a) and 8(b), we depict alignments of the (averaged)moments of the Dy and Fe atoms in Dy2Fe14B and those of the Ndand Fe atoms in Nd2Fe14B, respectively, as functions of hx .We noticed that the Dy and Fe moments did not yet becomeantiparallel and parallel, respectively, to the x-axis at hx ¼ 18:0 T,but they got antiparallel and parallel over hx ¼ 21:0 T, where Mz asa function of hx was nearly zero [Fig. 5(a)]. In that sense,hx ¼ 21:0 T is a critical value and may be adopted as hA. WhenhA ¼ 21:0 T was adopted, Msat ¼ Mz ¼ 0:96 μB=site at hz ¼ 21:0 T,which yielded K1 ≃ 6:88MJ=m3. These values for hA, Msat, and K1were also given by square brackets for reference in Table I.We compared our results with mean-field studies. Radwańskiand Franse studied magnetic properties of R2Fe14B using a simpletwo-sublattice (moments of R and Fe) mean-field theory.30They estimated the anisotropy constant of K1 treating only B00 forBml (¼ΘlAml hrli) for the crystal electric field energy of rare earthatoms. They gave K1 ¼ 21MJ/m3 at 300 K for Dy2Fe14B. Thisvalue is too large compared with the experimental value and ourTABLE I. Estimated values of hA, Msat, and K1 at T = 0.46Tc in comparison withexperimental data at room temperature. See text for values in square brackets.R2Fe14B hA (T) Msat (μB/site) K1 (MJ/m3)Dy2Fe14B (simulation) 15.0 [21.0] 0.85 [0.96] 4.35 [6.88](experiment) 15.0 0.824 4.22Nd2Fe14B (simulation) 6.0 1.65 3.31(experiment) 6.7 1.91 4.27FIG. 6. hx dependences of mz(Dy), mx (Dy), and m(Dy) in Dy2Fe14B in (a) alow field range and (b) in a wider field range. hx dependences of (c) mz(Fe),mx (Fe), and m(Fe) in Dy2Fe14B. T ¼ 0:46Tc.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917 136, 033904-6© Author(s) 2024 19 July 2024 00:02:16https://pubs.aip.org/aip/japestimation. This may be due to the ignorance of higher order of Bmland thermal fluctuation effects, which are adequately treated in theatomistic model approach besides the difference in modeling.Ito et al. also performed a simple two-sublattice mean-fieldstudy of the magnetic properties, treating B00, B20, and B40.31 Theyshowed under an applied field parallel to the hard axis (x direction)a small noncollinearity of Nd and Fe moments in Nd2Fe14B atroom temperature and a non collinearity of Dy and Fe moments inDy2Fe14B at a low temperature. Miura et al. studied the magneto-crystalline anisotropy in R2Fe14B in relation to non-collinearityeffect and showed that angular difference between the total and Femoments in Nd2Fe14B is very small in the whole temperatureregion, while a big angular difference exists in Dy2Fe14B especiallyat low temperatures.57We show in Fig. 8 that the average Nd and Fe moments havecollinearity at room temperature, which is close to Miura’s result.Our result also presented that the average Dy and Fe momentshave a slight non collinearity at room temperature in the fielddependence. This indicates that collinearity in Dy2Fe14B maybecome stronger at higher temperatures in the field dependence, inwhich temperature and thermal fluctuation effect may play animportant role.B. Effect of Dy random substitution on magnetizationreversal in the Nd magnetIn this subsection, we discuss the effects of Dy random substi-tution on the coercivity enhancement of Nd magnets. Because themodification of the (001) interface is more efficient for coercivityenhancement than that of the (100) interface,49 we focus on the(001) surface in the present work. We considered two types ofsystems. In system I, the (001) surface [top (n ¼ 1) and bottom(n ¼ 19)] layers were in contact with vacuum [Fig. 9(a)], whereasin system II, the surface layers were in contact with a soft magnet(grain boundary) phase [Fig. 9(b)]. The layers of Nd atoms in thehard magnet are numbered as n ¼ 1, 2, . . . in Figs. 9(a) and 9(b).We used open and periodic boundary conditions in systems I andII, respectively, along the c axis and periodic boundary conditionsalong the a and b axes in both systems. The hard (soft) magnetpart comprised 12� 12� 9 (12� 12� 3) unit cells along the a, b,and c axes.FIG. 7. hx dependences of (a) mz(Nd), mx (Nd), and m(Nd) and (b) mz(Fe),mx (Fe), and m(Fe) in Nd2Fe14B. T ¼ 0:46Tc.FIG. 8. Alignments of (a) Dy and Fe moments in Dy2Fe14B and (b) Nd and Femoments in Nd2Fe14B as a function of hx . T ¼ 0:46Tc.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917 136, 033904-7© Author(s) 2024 19 July 2024 00:02:16https://pubs.aip.org/aip/japThe structure of the grain boundary phase was complicated inthe experiments, and several amorphous-like structures, dependingon the experimental conditions, were suggested. It was difficult totheoretically estimate the microscopic parameters of the grain-boundary phase, and microscopic magnetic parameters are not cur-rently available. Only few first-principles studies on amorphous-likestructures have been conducted.58,59 We employed the same crystalstructure in the soft magnet phase as in the hard magnet phase butadopted smaller magnetic parameters, as has been applied inmicromagnetic simulations for Nd magnets. The values of allexchange interactions (anisotropy energies) inside the soft magnetphase were set to half (one-fifth) of those in the hard magnetphase. With this modeling, it is possible to catch important featuresof the effect of the soft magnet phase.In our previous study,50 all Nd atoms from the first to the nthlayer were substituted with Dy atoms, and n dependence of thethreshold field was investigated. In this study, Nd atoms were ran-domly substituted by Dy atoms with concentration P in the Ndlayers between n ¼ 1 and n ¼ 5 (between n ¼ 15 and n ¼ 19) insystems I and II [Figs. 9(a) and 9(b)], and P dependence of thethreshold field was investigated.In Fig. 10, P dependences of the threshold fields atT ¼ 0:46Tc for systems I and II are illustrated. The value (percent-age) of each symbol in the figure indicates the ratio of the thresholdfield to that of P ¼ 0. We observed that the threshold fieldsincreased almost linearly with P. The slope of the increase in thethreshold field in system II was nearly identical to that in systemI. In Ref. 50, we observed that n dependence of the threshold fieldswas approximately linear increase and the slope of the incrementwas nearly identical in both systems. This suggests that thelayer-by-layer substitution and random substitution caused no sig-nificant difference.When the concentration of Dy atoms was higher, the nucle-ation feature changed from surface nucleation to bulk nucleation,as in the case of n dependence of the threshold field in Ref. 50. Weanalyzed the critical value of P to distinguish between the surfaceand bulk nucleation. In Fig. 11, the ratios between the surface andbulk nucleation at the threshold fields are given as functions of Pin systems I and II at T ¼ 0:46Tc. We estimated the ratios from sixsamples of reversal at each P in each system. We found that alarger concentration (larger P) was necessary to realize the bulkFIG. 10. P dependences of the threshold fields at T ¼ 0:46Tc for systems I(dashed line with squares) and II (solid line with circles).FIG. 11. P dependences of the ratio between the surface and bulk nucleationat T ¼ 0:46Tc for systems I (dashed line with squares) and II (solid line withcircles).FIG. 9. (a) System I. The Nd surface layers are in contact with vacuum. (b)System II. The Nd surface layers are in contact with a soft magnet phase ofthree-unit-cell thickness. The Nd layers are numbered as n ¼ 1, 2, . . .. Nd(red) sites are replaced randomly by Dy atoms (orange) with concentration Pbetween n ¼ 1 and n ¼ 5 and between n ¼ 15 and n ¼ 19.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917 136, 033904-8© Author(s) 2024 19 July 2024 00:02:16https://pubs.aip.org/aip/japnucleation in system II than in system I, because a domain wallgenerated in the soft magnet phase caused a surface nucleationmore easily. The critical P in system I was Pc ≃ 40% and that insystem II was Pc ≃ 75%.In the study of n dependence, the critical n between thesurface and bulk nucleation was nc ≃ 1:5 for a system in contactwith vacuum and nc ≃ 3 for a system in contact with a soft magnetphase, corresponding to 30% and 60% Dy substitution betweenn ¼ 1 and n ¼ 5, respectively. Thus, the Dy substitution ratio forbulk nucleation was larger for random substitution than forlayer-by-layer substitution. However, the enhancement effectagainst the Dy substitution ratio was almost the same for thelayer-by-layer and random substitutions.These observations suggest that in Dy substitution near thesurface of the hard magnetic phase, the arrangement of Dy atomsis not very important, but the concentration density of Dy atoms issignificant for the enhancement of the coercivity of the Nd magnet,and Dy substitution enhances the coercive force with a ratio pro-portional to the density of Dy atoms.We compared our observation with a micromagnetics modelstudy.26 Bance et al. studied a micromagnetics model (50 nm) con-sisting of Nd2Fe14B core, 4-nm (Dy47,Nd53)2Fe14B shell, and a2-nm soft surface defect.26 They showed that the coercive force wasenhanced from 3.23 to 4.97 T at 300 K. This suggest 54% increaseof the coercive force with 47% Dy concentration for n ¼ 1 ton ≃ 7. In Fig. 10, the coercive force was enhanced from 3.10 to4.10 T at 0.46Tc indicating that 28% increase with 50% concentra-tion for n ¼ 1 to n ¼ 5. We confirmed a similar tendency of coer-civity enhancement between the two approaches (micromagneticand atomistic simulations), but found that the increase ratio of thecoercivity in our results is smaller than that of the micromagneticsimulations. This may be due to the inclusion of thermal fluctua-tion effects in the evaluation.V. SUMMARYTo understand the Dy substitution effect in neodymiummagnets, it is important to determine the differences in thermo-dynamic properties between Dy2Fe14B and Nd2Fe14B. In thispaper, we first investigated these differences using an atomisticmodel. Applying the Metropolis importance-sampling MonteCarlo (MC) method, we presented the temperature dependencesof the magnetizations of the magnets and their components atzero field. The simulated temperature-magnetization curve (Mz)of Dy2Fe14B exhibited characteristic features of the experimen-tally observed ones: Mz increased by lowering the temperaturefrom Tc, showed a peak at a high temperature, and slowlydecreased to zero temperature. This was found to originate fromthe Fe positive magnetization (mz) showing an upward convexdependence with a large curvature below Tc and the Dy negativemagnetization (mz) showing a downward-upward-downwardconvex dependence with low curvatures. On the other hand, thetemperature dependence of Mz on Nd2Fe14B was mainlyreflected by mz of the Fe moments. Although the origin of thespin reorientation (SR) is the property of the CEF potential ofNd atoms, mz of Fe moments was more reduced under T ¼ TRthan that of Nd atoms.The estimated value of the magnetization (Mz) of Dy2Fe14B atzero temperature agreed with the experimentally observed ones,and Mz at the peak position for Dy2Fe14B was also found to be anapproximation of the experimentally observed ones as well as agood approximation of the peak magnetization at the spin-reorientation transition temperature in Nd2Fe14B. We also foundthat the critical temperatures of the two magnets were very close,which was consistent with the experimental results, although thecritical temperatures were a little overestimated. This overestimationwas mainly due to an overestimation of the exchange interactions(Fe–Fe and Fe–R).The magnetization profile (mz , mxy at zero field) of Fe atomsin Dy2Fe14B was similar to that in Nd2Fe14B, except at tempera-tures below the SR transition temperature of Nd2Fe14B, which sug-gests that Fe moments are hardly affected by replacement of Nd byDy under zero field.We also studied the field dependences of the magnetizationsof the two magnets and their components. In Dy2Fe14B, Mx as afunction of hx increased linearly up to around 18T and then gradu-ally increased at higher fields. This situation makes an accurate esti-mation of the anisotropy field hA difficult. When the experimentalprocedure was followed in the simulation, the estimation corre-sponded to the experimental one: hA ¼ 15:0 T. We also estimatedhA with a different definition, in which the Dy and Fe momentswere antiparallel and parallel to the x axis (hard axis), respectively,and obtained hA ¼ 21:0 T. With increasing hx , the Dy and Femoments were rotated slightly tilting from the antiparallel configu-ration before 21.0 T. At higher hx , the Dy moment decreased(mx , 0) and increased (mx . 0) through a paramagnetic-likestate mx ≃ 0 at hx ≃ 98 T. On the other hand, in Nd2Fe14B, the Ndand Fe moments maintained collinearity (parallel configuration)under the field change of hx , and hA was estimated to be 6:0 Tclose to the experimental estimation: 6:7 T. We showed thatDy2Fe14B had much higher hA than Nd2Fe14B, as observed in theexperiments.We estimated the anisotropy energy K1 of Dy2Fe14B andNd2Fe14B from the obtained anisotropy fields (hA) and saturatedmagnetizations (Msat). When the experimental procedure was fol-lowed in the simulation, K1 ¼ 4:35MJ=m3 was obtained, whichwas close to the experimentally observed value K1 ¼ 4:22MJ=m3.However, using hA ¼ 21:0 T, K1 ¼ 6:88MJ=m3 was yielded. InNd2Fe14B, K1 ¼ 3:31MJ=m3 was obtained, which is an approxima-tion of the experimentally observed one: 4.27MJ/m3.Subsequently, we investigated the effect of Dy random substi-tution in a range close to the grain surface on the magnetizationreversal in the neodymium permanent magnet. We studied twosystems: system I whose surface layers were in contact with vacuumand system II whose surface layers were in contact with a softmagnet (grain boundary). The threshold fields increased almostlinearly with the concentration of Dy atoms, and the slopes of theincrements of the threshold fields in systems I and II were nearlyidentical. These results are similar to those obtained for n (layernumber) dependences of the threshold fields in the layer-by-layersubstitution studied in Ref. 50, although the critical densitybetween the surface and bulk nucleation was different. This indi-cates that if the Dy substitution is limited to the range close to thegrain surface, the arrangement of Dy atoms is not so important,Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917 136, 033904-9© Author(s) 2024 19 July 2024 00:02:16https://pubs.aip.org/aip/japbut the concentration density of Dy atoms is significant for theenhancement of the coercivity of the Nd magnet.ACKNOWLEDGMENTSThe authors would like to thank Dr. Hirosawa for instructivediscussions on the experimental results for R2Fe14B, Professor Akaifor helpful discussions on the magnetic parameters for Dy2Fe14Busing the KKR first-principles method, and Dr. Toga for useful dis-cussions on the properties of Nd magnets. This work was sup-ported by World Premier International Research Center Initiative(WPI), MEXT, Japan, and partially supported by Grants-in-Aid forScientific Research B (No. 24K01332) from MEXT. Numerical cal-culations were performed using the Numerical Materials Simulatorat the National Institute for Materials Science.AUTHOR DECLARATIONSConflict of InterestThe authors have no conflicts to disclose.Author ContributionsMasamichi Nishino: Conceptualization (lead); Data curation(equal); Formal analysis (lead); Funding acquisition (equal);Investigation (lead); Methodology (equal). Hiroshi Hayasaka: Datacuration (equal); Formal analysis (equal); Investigation (equal).Seiji Miyashita: Conceptualization (equal); Formal analysis(equal); Funding acquisition (equal); Investigation (equal).DATA AVAILABILITYThe data that support the findings of this study are availablefrom the corresponding author upon reasonable request.REFERENCES1R. Friedberg and D. I. Paul, “New theory of coercive force of ferromagneticmaterials,” Phys. Rev. Lett. 34, 1234–1237 (1975).2A. Sakuma, S. Tanigawa, and M. Tokunaga, “Micromagnetic studies of inhomo-geneous nucleation in hard magnets,” J. Magn. Magn. Mater. 84, 52–58 (1990).3A. Sakuma, “The theory of inhomogeneous nucleation in uniaxial ferromag-nets,” J. Magn. Magn. Mater. 88, 369–375 (1990).4S. Mohakud, S. Andraus, M. Nishino, A. Sakuma, and S. Miyashita,“Temperature dependence of the threshold magnetic field for nucleation anddomain wall propagation in an inhomogeneous structure with grain boundary,”Phys. Rev. B 94, 054430 (2016).5S. Hirosawa, M. Nishino, and S. Miyashita, “Perspectives for high-performancepermanent magnets: Applications, coercivity, and new materials,” Adv. Nat. Sci.:Nanosci. Nanotechnol. 8, 013002 (2017).6A. L. Wysocki and V. P. Antropov, “Micromagnetic simulations with periodicboundary conditions: Hard-soft nanocomposites,” J. Magn. Magn. Mater. 428,274–286 (2017).7I. E. Uysal, M. Nishino, and S. Miyashita, “Magnetic field threshold for nucle-ation and depinning of domain walls in the neodymium permanent magnetNd2Fe14B,” Phys. Rev. B 101, 094421 (2020).8S. Okamoto, R. Goto, N. Kikuchi, O. Kitakami, T. Akiya, H. Sepehri-Amin,T. Ohkubo, K. Hono, K. Hioki, and A. Hattori, “Temperature-dependent magne-tization reversal process and coercivity mechanism in Nd-Fe-B hot-deformedmagnets,” J. Appl. Phys. 118, 223903 (2015).9T. Pramanik, A. Roy, R. Dey, A. Rai, S. Guchhait, H. C. Movva, C.-C. Hsieh,and S. K. Banerjee, “Angular dependence of magnetization reversal in epitaxialchromium telluride thin films with perpendicular magnetic anisotropy,”J. Magn. Magn. Mater. 437, 72–77 (2017).10M. Sagawa and S. Hirosawa, “Magnetic hardening mechanism in sintered R–Fe–B permanent magnets,” J. Mater. Res. 3, 45–54 (1988).11J. F. Herbst, J. J. Croat, F. E. Pinkerton, and W. B. Yelon, “Relationshipsbetween crystal structure and magnetic properties in Nd2Fe14B,” Phys. Rev. B 29,4176–4178 (1984).12S. Hirosawa, Y. Matsuura, H. Yamamoto, S. Fujimura, M. Sagawa, andH. Yamauchi, “Single crystal measurements of anisotropy constants of R2Fe14B(R=Y, Ce, Pr, Nd, Gd, Tb, Dy and Ho),” Jpn. J. Appl. Phys. 24, L803–L805 (1985).13A. V. Andreev, A. V. Deriagin, N. V. Kudrevatykh, N. V. Mushnikov, andV. A. Reimer, “The magnetism of Y2Fe14B and Nd2Fe14B and their hydrides,”Zh. Eksperimentalnoi Teor. Fiz. 90, 1042–1050 (1986).14H. Kronmüller, K.-D. Durst, and M. Sagawa, “Analysis of the magnetic hard-ening mechanism in RE-FeB permanent magnets,” J. Magn. Magn. Mater. 74,291–302 (1988).15J. F. Herbst, “R2Fe14B materials: Intrinsic properties and technologicalaspects,” Rev. Mod. Phys. 63, 819–898 (1991).16S. Hirosawa, Y. Matsuura, H. Yamamoto, S. Fujimura, M. Sagawa, andH. Yamauchi, “Magnetization and magnetic anisotropy of R2Fe14B measured onsingle crystals,” J. Appl. Phys. 59, 873–879 (1986).17O. Yamada, Y. Ohtsu, F. Ono, M. Sagawa, and S. Hirosawa,“Magnetocrystalline anisotropy in Nd2Fe14B intermetallic compound,” J. Magn.Magn. Mater. 70, 322–324 (1987).18N. V. Mushnikov, P. B. Terent’ev, and E. V. Rosenfel’d, “Magnetic anisotropyof the Nd2Fe14B compound and its hydride Nd2Fe14BH4,” Phys. MetalsMetallogr. 103, 39–50 (2007).19T. Kohashi, K. Motai, T. Nishiuchi, and S. Hirosawa, “Magnetism in grain-boundary phase of a NdFeB sintered magnet studied by spin-polarized scanningelectron microscopy,” Appl. Phys. Lett. 104, 232408 (2014).20S. Sugimoto, “Current status and recent topics of rare-earth permanentmagnets,” J. Phys. D: Appl. Phys. 44, 064001 (2011).21K. Hirota, H. Nakamura, T. Minowa, and M. Honshima, “Coercivity enhance-ment by the grain boundary diffusion process to Nd-Fe-B sintered magnets,”IEEE Trans. Magn. 42, 2909–2911 (2006).22F. Xu, J. Wang, X. Dong, L. Zhang, and J. Wu, “Grain boundary microstruc-ture in DyF3-diffusion processed Nd-Fe-B sintered magnets,” J. Alloys Compd.509, 7909–7914 (2011).23K. Löewe, C. Brombacher, M. Katter, and O. Gutfleisch,“Temperature-dependent Dy diffusion processes in Nd-Fe-B permanentmagnets,” Acta Mater. 83, 248–255 (2015).24W. Chen, J. M. Luo, Y. W. Guan, Y. L. Huang, M. Chen, and Y. H. Hou,“Grain boundary diffusion of Dy films prepared by magnetron sputtering forsintered Nd–Fe–B magnets,” J. Phys. D: Appl. Phys. 51, 185001 (2018).25T.-H. Kim, T. Sasaki, T. Ohkubo, Y. Takada, A. Kato, Y. Kaneko, andK. Hono, “Microstructure and coercivity of grain boundary diffusion processedDy-free and Dy-containing Nd–Fe–B sintered magnets,” Acta Mater. 172,139–149 (2019).26S. Bance, J. Fischbacher, A. Kovacs, H. Oezelt, F. Reichel, and T. Schrefl,“Thermal activation in permanent magnets,” JOM 67, 1350–1356 (2015).27J. Fischbacher, A. Kovacs, L. Exl, J. Kühnel, E. Mehofer, H. Sepehri-Amin,T. Ohkubo, K. Hono, and T. Schrefl, “Searching the weakest link: Demagnetizingfields and magnetization reversal in permanent magnets,” Scr. Mater. 154,253–258 (2018).28H. Kronmüller and M. Fähnle, Micromagnetism and the Microstructure ofFerromagnetic Solids, Cambridge Studies in Magnetism (Cambridge UniversityPress, 2003).29G. Grinstein and R. H. Koch, “Coarse graining in micromagnetics,” Phys. Rev.Lett. 90, 207201 (2003).30R. J. Radwański and J. J. M. Franse, “Rare-earth contribution to the magneto-crystalline anisotropy energy in R2Fe14B,” Phys. Rev. B 36, 8616–8621 (1987).Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917 136, 033904-10© Author(s) 2024 19 July 2024 00:02:16https://doi.org/10.1103/PhysRevLett.34.1234https://doi.org/10.1016/0304-8853(90)90162-Jhttps://doi.org/10.1016/0304-8853(90)90660-Ihttps://doi.org/10.1103/PhysRevB.94.054430https://doi.org/10.1088/2043-6254/aa597chttps://doi.org/10.1088/2043-6254/aa597chttps://doi.org/10.1016/j.jmmm.2016.11.128https://doi.org/10.1103/PhysRevB.101.094421https://doi.org/10.1063/1.4937274https://doi.org/10.1016/j.jmmm.2017.04.039https://doi.org/10.1557/JMR.1988.0045https://doi.org/10.1103/PhysRevB.29.4176https://doi.org/10.1143/JJAP.24.L803https://doi.org/10.1016/0304-8853(88)90202-8https://doi.org/10.1103/RevModPhys.63.819https://doi.org/10.1063/1.336611https://doi.org/10.1016/0304-8853(87)90456-2https://doi.org/10.1016/0304-8853(87)90456-2https://doi.org/10.1134/S0031918X0701005Xhttps://doi.org/10.1134/S0031918X0701005Xhttps://doi.org/10.1063/1.4883487https://doi.org/10.1088/0022-3727/44/6/064001https://doi.org/10.1109/TMAG.2006.879906https://doi.org/10.1016/j.jallcom.2011.05.023https://doi.org/10.1016/j.actamat.2014.09.039https://doi.org/10.1016/j.actamat.2019.04.032https://doi.org/10.1007/s11837-015-1415-7https://doi.org/10.1016/j.scriptamat.2017.11.020https://doi.org/10.1103/PhysRevLett.90.207201https://doi.org/10.1103/PhysRevLett.90.207201https://doi.org/10.1103/PhysRevB.36.8616https://pubs.aip.org/aip/jap31M. Ito, M. Yano, N. M. Dempsey, and D. Givord, “Calculations of the mag-netic properties of R2M14B intermetallic compounds (R=rare earth, M=Fe, Co),”J. Magn. Magn. Mater. 400, 379–383 (2016).32P. Peczak, A. M. Ferrenberg, and D. P. Landau, “High-accuracy Monte Carlostudy of the three-dimensional classical Heisenberg ferromagnet,” Phys. Rev. B43, 6087–6093 (1991).33J. L. García-Palacios and F. J. Lázaro, “Langevin-dynamics study of the dynam-ical properties of small magnetic particles,” Phys. Rev. B 58, 14937–14958(1998).34M. Nishino and S. Miyashita, “Realization of the thermal equilibrium in inho-mogeneous magnetic systems by the Landau-Lifshitz-Gilbert equation with sto-chastic noise, and its dynamical aspects,” Phys. Rev. B 91, 134411 (2015).35Y. Toga, M. Matsumoto, S. Miyashita, H. Akai, S. Doi, T. Miyake, andA. Sakuma, “Monte Carlo analysis for finite-temperature magnetism ofNd2Fe14B permanent magnet,” Phys. Rev. B 94, 174433 (2016).36M. Nishino, Y. Toga, S. Miyashita, H. Akai, A. Sakuma, and S. Hirosawa,“Atomistic-model study of temperature-dependent domain walls in the neodym-ium permanent magnet Nd2Fe14B,” Phys. Rev. B 95, 094429 (2017).37T. Hinokihara, M. Nishino, Y. Toga, and S. Miyashita, “Exploration of the effects ofdipole-dipole interactions in Nd2Fe14B thin films based on a stochastic cutoff methodwith a novel efficient algorithm,” Phys. Rev. B 97, 104427 (2018).38S. Miyashita, M. Nishino, Y. Toga, T. Hinokihara, T. Miyake, S. Hirosawa, andA. Sakuma, “Perspectives of stochastic micromagnetism of Nd2Fe14B and com-putation of thermally activated reversal process,” Scr. Mater. 154, 259–265(2018).39Y. Toga, M. Nishino, S. Miyashita, T. Miyake, and A. Sakuma, “Anisotropy ofexchange stiffness based on atomic-scale magnetic properties in the rare-earthpermanent magnet Nd2Fe14B,” Phys. Rev. B 98, 054418 (2018).40M. Nishino and S. Miyashita, “Nontrivial temperature dependence of ferro-magnetic resonance frequency for spin reorientation transitions,” Phys. Rev. B100, 020403 (2019).41Y. Toga, S. Miyashita, A. Sakuma, and T. Miyake, “Role of atomic-scalethermal fluctuations in the coercivity,” npj Comput. Mater. 6, 67 (2020).42M. Nishino, I. E. Uysal, T. Hinokihara, and S. Miyashita, “Dynamical aspectsof magnetization reversal in the neodymium permanent magnet by a stochasticLandau-Lifshitz-Gilbert simulation at finite temperature: Real-time dynamicsand quantitative estimation of coercive force,” Phys. Rev. B 102, 020413(R) (2020).43S. Westmoreland, R. Evans, G. Hrkac, T. Schrefl, G. Zimanyi, M. Winklhofer,N. Sakuma, M. Yano, A. Kato, T. Shoji, A. Manabe, M. Ito, and R. Chantrell,“Multiscale model approaches to the design of advanced permanent magnets,”Scr. Mater. 148, 56–62 (2018).44S. C. Westmoreland, C. Skelland, T. Shoji, M. Yano, A. Kato, M. Ito, G. Hrkac,T. Schrefl, R. F. L. Evans, and R. W. Chantrell, “Atomistic simulations of α-Fe/Nd2Fe14B magnetic core/shell nanocomposites with enhanced energy productfor high temperature permanent magnet applications,” J. Appl. Phys. 127,133901 (2020).45Q. Gong, M. Yi, R. F. L. Evans, B.-X. Xu, and O. Gutfleisch, “Calculatingtemperature-dependent properties of Nd2Fe14B permanent magnets by atomisticspin model simulations,” Phys. Rev. B 99, 214409 (2019).46Q. Gong, M. Yi, and B.-X. Xu, “Multiscale simulations toward calculatingcoercivity of Nd–Fe–B permanent magnets at high temperatures,” Phys. Rev.Mater. 3, 084406 (2019).47Q. Gong, M. Yi, R. F. L. Evans, O. Gutfleisch, and B.-X. Xu, “Anisotropicexchange in Nd–Fe–B permanent magnets,” Mater. Res. Lett. 8, 89–96 (2020).48S. Miyashita, M. Nishino, Y. Toga, T. Hinokihara, I. E. Uysal, T. Miyake,H. Akai, S. Hirosawa, and A. Sakuma, “Atomistic theory of thermally activatedmagnetization processes in Nd2Fe14B permanent magnet,” Sci. Technol. Adv.Mater. 22, 658–682 (2021).49M. Nishino, I. E. Uysal, and S. Miyashita, “Effect of the surface magneticanisotropy of neodymium atoms on the coercivity in neodymium permanentmagnets,” Phys. Rev. B 103, 014418 (2021).50M. Nishino, H. Hayasaka, and S. Miyashita, “Microscopic origin of coercivityenhancement by dysprosium substitution into neodymium permanent magnets,”Phys. Rev. B 106, 054422 (2022).51H. Hayasaka, M. Nishino, and S. Miyashita, “Microscopic study on theangular dependence of coercivity at zero and finite temperatures,” Phys. Rev. B105, 224414 (2022).52M. Nishino and S. Miyashita, “Quantitative estimation of coercive field in aferromagnetic grain using field sweep simulation,” Phys. Rev. B 107, 184422(2023).53R. J. Elliott and K. W. H. Stevens, “The theory of magnetic resonance experi-ments on salts of the rare earths,” Proc. R. Soc. A 218, 553–566 (1953).54Y. Miura, H. Tsuchiura, and T. Yoshioka, “Magnetocrystalline anisotropy ofthe Fe-sublattice in Y2Fe14B systems,” J. Appl. Phys. 115, 17A765 (2014).55M. Yamada, H. Kato, H. Yamamoto, and Y. Nakagawa, “Crystal-field analysisof the magnetization process in a series of Nd2Fe14B-type compounds,” Phys.Rev. B 38, 620–633 (1988).56A. J. Freeman and R. E. Watson, “Theoretical investigation of some magneticand spectroscopic properties of rare-earth ions,” Phys. Rev. 127, 2058–2075(1962).57D. Miura and A. Sakuma, “Non-collinearity effects on magnetocrystallineanisotropy for R2Fe14B magnets,” J. Phys. Soc. Jpn. 88, 044804 (2019).58Y. Tatetsu, S. Tsuneyuki, and Y. Gohda, “First-principles study of the role ofCu in improving the coercivity of Nd-Fe-B permanent magnets,” Phys. Rev.Appl. 6, 064029 (2016).59Y. Gohda, Y. Tatetsu, and S. Tsuneyuki, “Electron theory on grain-boundarystructures and local magnetic properties of neodymium magnets,” Mater. Trans.59, 332–337 (2018).Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 136, 033904 (2024); doi: 10.1063/5.0217917 136, 033904-11© Author(s) 2024 19 July 2024 00:02:16https://doi.org/10.1016/j.jmmm.2015.08.065https://doi.org/10.1103/PhysRevB.43.6087https://doi.org/10.1103/PhysRevB.58.14937https://doi.org/10.1103/PhysRevB.91.134411https://doi.org/10.1103/PhysRevB.94.174433https://doi.org/10.1103/PhysRevB.95.094429https://doi.org/10.1103/PhysRevB.97.104427https://doi.org/10.1016/j.scriptamat.2017.11.012https://doi.org/10.1103/PhysRevB.98.054418https://doi.org/10.1103/PhysRevB.100.020403https://doi.org/10.1038/s41524-020-0325-6https://doi.org/10.1103/PhysRevB.102.020413https://doi.org/10.1016/j.scriptamat.2018.01.019https://doi.org/10.1063/1.5126327https://doi.org/10.1103/PhysRevB.99.214409https://doi.org/10.1103/PhysRevMaterials.3.084406https://doi.org/10.1103/PhysRevMaterials.3.084406https://doi.org/10.1080/21663831.2019.1702116https://doi.org/10.1080/14686996.2021.1942197https://doi.org/10.1080/14686996.2021.1942197https://doi.org/10.1103/PhysRevB.103.014418https://doi.org/10.1103/PhysRevB.106.054422https://doi.org/10.1103/PhysRevB.105.224414https://doi.org/10.1103/PhysRevB.107.184422https://doi.org/10.1063/1.4869061https://doi.org/10.1103/PhysRevB.38.620https://doi.org/10.1103/PhysRevB.38.620https://doi.org/10.1103/PhysRev.127.2058https://doi.org/10.7566/JPSJ.88.044804https://doi.org/10.1103/PhysRevApplied.6.064029https://doi.org/10.1103/PhysRevApplied.6.064029https://doi.org/10.2320/matertrans.M2017258https://pubs.aip.org/aip/jap