arXiv:1804.05824v2 [cond-mat.mtrl-sci] 24 Jul 2018 srqie o nutilapiain[]bcuei a a has ( it temperature because Curie [6] application low relatively industrial temperature for high required at is at of permanent improvement [3–5], performance RT highest the as known inmtl 1 ] oee,i Nd transi- in the However, of interac- order 2]. an ferromagnetic [1, metals by strong tion (RT) the with temperature tion room orig- at anisotropy 4 maintained magnetic the ferro- to strong from strong scale The inated and atomic anisotropy order. in magnetic magnetic combine strong metals produce transition the and SmCo ri hss[,8.Frti esn td ftecoer- the ( of scale atomic study the reason, from this analyses For and requires main civity 8]. the of of [7, properties phases microstructure magnetic grain and the boundary on energy also grain anisotropy but the phase the main as the intrin- such of the parameters, only not magnetic on sic depends coercivity the Therefore, . rare-earth other the with ei rnnmgei ri onaypae ( phases boundary grain non-magnetic or netic ftemi aeerhmge hs ( phase magnet rare-earth composed main materials the polycrystalline of are use practical the h arsoi cl ( scale macroscopic the in na tmcsae( scale atomic an fluctua- in tions thermal and the structures consider magnetic should inhomogeneous we etc.), temperature, exchange Curie anisotropy, mag- stiffness, magnetic rare-earth , the ana- (e.g., of To nets properties yet. magnetic elucidated fully intrinsic been lyze not has magnets earth lcdt h oriiyi ytmtcmanner. systematic a in coercivity the elucidate nrr-at emnn ant uha Nd as such magnets permanent rare-earth In oriiymcaimo Nd of mechanism Coercivity 5 Sm , nstoyo xhnesins ae naoi-cl magn atomic-scale on based stiffness exchange of Anisotropy uaToga, Yuta 4 DFa,Ntoa nttt fAvne nutilScienc Industrial Advanced of Institute National CD-FMat, 2 aetmge,Nd magnet, manent al M)mto iha tmsi pnmdladLandau-Li and simulati MC model atomistic spin The Nd atomistic model. of magnetic an continuous with a with method (MC) Carlo efre ihteprmtr bandb h Csimulation MC of the property by thermal obtained the parameters and the with performed h aee tutr fN tm n ekecag coupling exchange weak of anisotropy and the atoms that Nd confirm of also structure layered the eesl(oriiy ie ytedpnigprocess. depinning the by given (coercivity) reversal Fe eeaieteaiorpcpoete fteecag stiffn exchange the of properties anisotropic the examine We 17 .INTRODUCTION I. N 2 f Fe 3 n oo,terr-at elements rare-earth the on, so and , eetoso h aeerhao is atom rare-earth the of -electrons 1 14 1 aaih Nishino, Masamichi SCM ainlIsiuefrMtrasSine(IS,T (NIMS), Science Materials for Institute National ESICMM, osrce rma-ntocluain,adteLGmicr LLG the and calculations, ab-initio from constructed B ∼ ∼ µ 5 2 ) h aeerhmgesin magnets rare-earth The A). ˚ ) hsfc a niie to inhibited has fact This m). ainlIsiuefrMtrasSine(IS,Tsukuba, (NIMS), Science Materials for Institute National eateto ple hsc,Thk nvriy Sendai, University, Tohoku Physics, Applied of Department eerhCne o dacdMaueetadCharacteriza and Measurement Advanced for Center Research 3 eateto hsc,TeUiest fTko oy,Japa Tokyo, Tokyo, of University The Physics, of Department nrr-at emnn antNd magnet permanent rare-earth in 2 Fe 2 Fe 14 ∼ 14 2 n h te rare- other the and B ,b oncigaaye ihtodffrn clso length of scales different two with analyses connecting by B, Fe 8 )a compared as K) 585 14 ∼ A hc swell is which B µ ,1 2, ) n mag- and m), eedo h retto ntecytl hc r attribut are which crystal, the in orientation the on depend ej Miyashita, Seiji A Dtd uy2,2018) 25, July (Dated: ∼ infiatyaet h hehl edfrtemagnetizati the for field threshold the affects significantly ∼ 2 Fe )to A) ˚ nm). 14 B, ftetmeauedpnec sfra eknow. we as far as estimation dependence theoretical temperature no the is of there temper- also several and at [19–22], only atures observed are values the mentally, niae sn biii aclto that calculation ab-initio using indicated lu ie[] o oeohrmgei aeil,YCo materials, magnetic other nu- some reversal) and For (magnetization (DW) [7]. size critical wall cleus domain the the and to related [23] width is quantity temper- This each at ature. magnets continuous of stiffness exchange a n hoeia aesuidtetemlpoete of properties thermal the K studied have theoretical and tal otmgei hss[–2,i a enidctdthat indicated been and has hard it of both [9–12], consisting phases systems magnetic junction the soft In (2)). Eq. and n mrv h oriiya ihtmeaue,i is it of dependences temperatures, temperature high mechanism the at coercivity clarify to re- the coercivity important uniform elucidate the the to improve of order and value In the from versal. which coercivity depinning) the and reduce nucleation (i.e., reversals tization h retto ntecytl 2] n uaaae al. et Fukazawa anisotropic And the out [24]. pointed crystals also the in orientation the encridotwt otnu oe htue the uses that model constants, continuum anisotropy a with magnetic out carried been tffesconstant, stiffness a,mcocpcudrtnigo h eprtr and temperature the of understanding microscopic way, anisotropic of effect the [24], A depen- Ref. orientation in an Although, has [26]. also dence it calcula- that ab-initio indicated from and tions, constructed model spin atomistic eauedpnec fD it fNd tem- of the width examined DW of al. dependence et perature Nishino Recently, [25]. pounds ,1 3, u h au of value The hoeial nlssfrtececvt nms have most in coercivity the for analyses Theoretically ncecvt sas icse naphenomenological a in discussed also is coercivity on 1–8.Hwvr regarding However, [13–18]. L aah Miyake, Takashi A K 1 n ehooy(IT,Tuua Japan Tsukuba, (AIST), Technology and e hrfr,mn eerhr nbt experimen- both in researchers many Therefore, . 0 u tp ant CP,FP,FP) Belashchenko FePt), FePd, (CoPt, magnets -type and n r efre ntesi model spin the on performed are ons s constant, ess ewe dadF tm.We atoms. Fe and Nd between s .W lrf htteamplitude the that clarify We s. A A sizGlet(L)equation (LLG) fshitz-Gilbert aealreeeto noeetmagne- incoherent on effect large a have sgvna arsoi rprisof properties macroscopic a as given is 2 mgeissmltosare simulations omagnetics Fe A ,1 4, uua Japan sukuba, 14 smcocpcprmtr (see parameters macroscopic as A Japan n kms Sakuma Akimasa and B Japan o aeerhper- rare-earth for , tion, n tcproperties etic A A .. Monte i.e., , K o mF,Co) Sm(Fe, for o Nd for u n h exchange the and , dto ed 2 2 Fe on Fe A 14 14 eed on depends sn an using B 5 ,experi- B, 12 com- K u 5 2 orientation dependences of A is essential for the coerciv- ies [18, 26, 27]. Exchange coupling constants, J˜ij , ity mechanism in Nd2Fe14B magnet. were calculated with Liechtenstein’s formula [36] on the In the present paper, by a comparison of the results Korringa-Kohn-Rostoker (KKR) Green’s-function code, obtained by a continuum model and those obtained by machikaneyama (akaikkr) [37]. In the present study, the atomistic spin model constructed from ab-initio cal- to reduce computational cost, we use only short-range culations (same spin model as Ref. [18, 26, 27]), we study exchange couplings within the range of rcut = 3.52 A,˚ the temperature and the orientation dependence of A for in which primary Fe–Fe and Nd–Fe interactions are in- ml Nd2Fe14B magnet. The comparison of the two differ- cluded. Anisotropy terms, Di and Bl,i , were determined ent scale models is done by making use of magnetic DW from the previous first-principles calculation [38] and the energy. This scheme has been used successful for FePt experimental result [35], respectively. Consequently, the [28–30] and Co [31] magnetic materials. We find that atomistic spin model uses many input parameters in a A has the anisotropic property in Nd2Fe14B. Moreover, unit cell which includes 68 atoms (see Fig. 1 (a)). The the reason for the anisotropy is attributed to the weak previous study [18] confirmed that the model and param- exchange couplings of Nd and Fe atoms. Additionally, eters are highly reliable for the magnetic properties of the we performed micromagnetics simulations on the config- Nd2Fe14B magnet. uration in which a soft magnetic phase is attached to (001) plane or (100) plane of a grain of Nd2Fe14B. The simulations predict that the anisotropy of A reduces the B. Continuum Model coercivity for the former configuration because A of z- direction is larger than that of x-direction, while it en- hances the coercivity for the latter configuration. The Under the continuum approximation, the micromag- series of the calculation schemes in the present study cor- netic energy of the exchange couplings and the magnetic responds to the multiscale analysis [32, 33] that connects anisotropies at temperature T is expressed as follows [39]: different scales from ab-initio calculation to coercivity as macroscopic physics. 3 2 Econt = d r Al(T ) (∇lm(r)) V   Z l=Xx,y,z II. MODELS AND METHOD 3  + d r EK (T,θ(r)), (2) A. Atomistic Spin Model ZV where m denotes a normalized magnetization vector, To include the information of electronic states at an Al(T ) is the exchange stiffness constant of each direction atomistic scale, we treat the (spin) magnetic moment of (x,y,z) in the crystal, and EK is the each atom as a classical spin and then construct a clas- energy which is usually expressed as: sical Heisenberg model. The atomistic Hamiltonian has 2 4 6 the form: EK (T,θ)= K1(T ) sin θ + K2(T ) sin θ + K4(T ) sin θ, (3) H = −2 J˜ij ei · ej i

pressed by the following formula: ∞ E (T ′) F (T )= T dT ′ dw . (5) dw (T ′)2 ZT The DW internal energy, Edw(T ), is defined as an en- ergy difference between the internal energies obtained in different boundary conditions with and without DW. For this calculation, we fix the boundaries in the direction an- tiparallel (AP) or parallel (PA). Additionally, since the crystal structure of Nd2Fe14B has anisotropy, we con- sider two models with the two fixed boundaries depicted in Fig. 1(b) and (c). Thus, we have two types of DWs [26]. One is a DW along the yz-plane (type I), and the other is along the xy-plane (type II). We set the periodic bound- ary condition in the yz(xy)-plane for the model of type I (type II). To calculate EK (T,θ), we adopt the constrained MC (C-MC) method for the atomistic spin model. The C-MC method samples spin configurations under the condition of a fixed angle of total magnetization, which allows us to calculate the angle dependence of the spin torque and the free energy [42]. By considering that EK is equal to the free energy of magnetic anisotropy, we calculate EK from the atomistic model. The previous study [18] showed the validity of the C-MC methods for calculating Ku(T ) in EK (T,θ) of the Nd2Fe14B atomistic spin model. Note 2 that if Eq. (3) is given in the form EK = K1 sin θ (K2 = K4 = 0), Eq. (4) can be integrated analytically and gives the well known relation: Edw(T )=4 A(T )K1(T ). In the present paper, the MC and C-MC simulations run 100000-200000 MC steps for equilibriump and the fol- lowing 100000-200000 MC steps for taking statistical av- erages, where a MC step corresponds to one trial for each FIG. 1. (a) Unit Cell of Nd2Fe14B including 68 atoms [5]. spin to be updated. We calculated the average of mag- Two types of spin configurations for Nd2Fe14B atomistic spin model: DW orientation is along (b) the yz-plane (TypeI) and netic anisotropy energies and DW energies from 8-20 dif- (c) the xy-plane (Type II). The arrows in figures describe the ferent runs with different initial conditions and random antiparallel (AP) boundary condition. These crystal struc- sequences. tures were plotted using vesta [41].

III. RESULTS AND DISCUSSION

C. Methods for determining Ku(T ) and A(T ) A. Magnetic anisotropy and domain wall energy In the present study, the macroscopic parameters are evaluated by comparing the energies of the DW and of As mentioned above, the evaluation of the exchange the magnetic anisotropy obtained by the above two mod- stiffness constant requires precise calculations of the mag- els at each temperature. We use a Monte Carlo (MC) netic anisotropy constants and the DW energy. First, scheme to calculate Helmholtz free energies on the atom- we show the temperature dependence of the magnetic istic spin model. In order to evaluate A, we consider the anisotropy constants calculated by the C-MC method in two quantities obtained from DW energy: Edw(T ) and Fig. 2, which qualitatively agrees with previous experi- ments [14, 15]. There, we used the system of 10 × 10 × 7 Fdw(T ) [23, 29]. Edw is the DW energy in the continuum model, which is expressed by the following formula: unit cells imposing the periodic boundary conditions. In uniaxial anisotropy, we study the quantities: FA(T ) ≡ π EK (T,θ = π/2) = EK (T,π/2) −EK (T, 0) = u Ku(T ). Edw(T )=2 A(T ) dθ EK (T,θ). (4) Figure 2 shows that FA(T ) rapidly decreases with the 0 P p Z p temperature below 400K. This temperature dependence To obtain A(T ), we need the values of Edw(T ) and is understood as a result of fragile thermal properties EK (T,θ). We regard Edw(T ) to be equal to the DW free of Nd atoms. That is, while at low temperature, the energy in the atomistic spin model, Fdw(T ), which is ex- anisotropy of Nd atoms is dominant, the anisotropy of 4

π E FA(T ) R0 dθp K (T, θ) ) ) 3 3 10 0.4 Lx m m

/ 1.00 1.00 / 6 8

(MJ 0.2 10 A ∞ 0.96 0.98 F 5 0.0 700 800 900 Temp. (K) 0.92 0.96 200 K 300 K 0.88 0.94 400 K 0 500 K K1 K2 K4 FA 600 K 0.84 0.92 700 K Magnetic anisotropy (MJ 800 K 0 200 400 600 800 1000 Temp. (K) 0.00 0.02 0.04 0.06 0.00 0.02 0.04 0.06 − 1 − 1 N 3 N 3

FIG. 2. Magnetic anisotropy constants and FA as a function F of temperature for the system of 10 × 10 × 7 unit cells. In- FIG. 3. Finite size extrapolation of A(T ) and π E set show FA(T ) for each system size, (Lx, Ly, Lz) = (6, 6, 4), R0 dθp K (T,θ). Solid lines are linear fits to the Monte Carlo 1/3 results for each system size (N is the total number of spins). (8, 8, 6), (10, 10, 7), and the extrapolation data (Lx ∝ N → ∞). Vertical axes are normalized at the values for N = 3264.

Edw. To analyze in detail the temperature dependence, Nd atoms decreases with the temperature more rapidly K we divided Edw into the magnetic anisotropy term (Edw) than that of Fe atoms, because the exchange field of a J and the exchange term (Edw), and plot the contributions Nd atom (sum of exchange couplings that connect to a in Fig. 4. The DW energies of typeI and typeII show a Nd atom) is much smaller (about 20-25 %) than that of qualitatively similar temperature dependence. Both en- a Fe atom (see also the detailed discussion in Ref. [18]). ergy terms naturally vanish for T ≥ TC, because DW Additionally, owing to the higher order terms of the does not appear even in the AP boundary condition. For 0 0 Nd anisotropy, B4 ,B6 , the magnetization of Nd2Fe14B T

(a) TypeI (yz-plane)

(b) TypeII (xy-plane) FIG. 5. Temperature dependence of DW free energy, Fdw and internal energy, Edw (same as Fig. 4 (a) and (b)), for each DW type.

10 Exp. m)

15 / (pJ ˜ A 0

m) 10 0 0.5 1 / Temp./TC (pJ A 5 Ax (TypeI)

Az (TypeII) 0 0 200 400 600 800 1000 FIG. 4. Temperature dependence of DW internal energy for Temp. (K) three system sizes, (Lx, Ly, Lz), in the DWs of (a) Type I and (b) Type II. Total DW energy is divided into anisotropy and K J exchange terms, Edw = Edw +Edw, for the largest system size of each DW type. FIG. 6. Temperature dependence of exchange stiffness con- stant, A for each DW type. Solid (dotted) lines are fitting results in the range of 200-860 K (200-400 K) using A(T ) = the thermal fluctuation. As a thermally averaged mag- CM(T )n. Inset show the renormalized values, A˜ (see Eq. 6), netic structure, the non-collinear structure (the spiral and the green bar denotes the range of the experimental val- structure) becomes a collinear structure. At this point, ues at room temperature [21, 22]. J Edw takes the peak value at Th and then decreases rapidly and approaches zero towards TC. This break of the non-collinear structure was discussed in previous stud- versal starts from the z-plane. Moreover, the generation ies as the disappearance of an xy–magnetization inside of the DW in the magnets would also depend on the the DW [29, 44, 45]. grain boundary phase and the dipole-dipole interaction. By those internal DW energies and Eq. (5), we can We will discuss these properties in more realistic magne- tization reversal process in Sec. III C. calculate DW free energies, Fdw. In Fig. 5, we plot the temperature dependence of Fdw and Edw (the same data shown in Fig. 4) for both DW types. The differences B. Exchange stiffness constant between Fdw and Edw correspond to the contribution of magnetic entropy. The difference in the DW energy be- tween type I and type II naturally indicates an anisotropy The exchange stiffness constants, A, for the two direc- concerning to the direction in which DWs are generated. tions can be evaluated by using Eq. (4) and the numeri- The DW prefers to be generated in the configuration of cal results for the magnetic anisotropy and the DW en- type II. These observations imply the magnetization re- ergy, whose temperature dependence are shown in Fig. 6. 6

the periodic boundary condition in the yz(xy)-plane for 12 Type I the model of type I (type II), whereas the previous cal- Type II culation was performed under the open-boundary condi- 10 tions for both models. Our results of δdw qualitatively agree with the previous results, although they tend to 8 take a smaller value because the thermal fluctuation be- comes smaller than the previous study due to the differ- (nm) 6 ence of boundary conditions. The comparisons with the dw δ previous experiments and the numerical study as men- 4 tioned above guarantee our results concerning to A. 2 Now, we study its thermal properties. The temper- ature dependence of A is often discussed in relation to 0 magnetization by using the following expression: 0 200 400 600 800 1000 Temp. (K) M(T ) n A(T )= A(0) , (8) M(0)   FIG. 7. Temperature dependence of DW width, δdw, for each where M(T ) is the amplitude of the magnetization (not DW type. Circle lines are our calculation results from Ku(T ) shown, see Ref. [18]). Under a mean field approxima- F and dw. Square points are the previous numerical results tion with homogeneous spin systems, the exponent n which are evaluated directly from the snapshots of spin con- is 2 [30, 46]. However, under more accurate methods, figurations while in MC simulations. the exponent takes a value different from 2, for exam- ple, FePt: n = 1.76 [30], hcp-Co: n = 1.79 [31]. Here, we define the value of A calculated from the con- Thus we estimate n to examine the thermal properties figuration of type I (type II) as the exchange stiffness for the Nd2Fe14B, by fitting A(T ) in Fig. 6 with the n constant of the x (z)-direction, Ax(z). Reflecting the form CM(T ) , where C is a fitting constant. Note anisotropy of the DW energy, the exchange stiffness con- that n depends on the fitting range of temperature stant naturally has the anisotropy depending on the di- nx(z) = 1.68(1.84) in the range of 200-860K, whereas rection in the crystal. For the comparison of A with ex- nx(z) = 1.46(1.69) in the range of 200-400K. Fitting perimental values, it is reasonable to normalize the tem- lines of the former and the latter are plotted by the solid perature dependence with TC, because the spin model and dotted lines in Fig. 6, respectively. Microscopically, MC Fe and Nd show different thermal properties. Indeed, Nd overestimates (TC = 870K) compared to experiment EXP atoms have weak exchange coupling and so do not have (TC ≃ 585 K). In addition, as pointed in the mean field approximation [46], A is roughly proportional to TC, much influence on A, whereas they have a large magnetic so that we also rescale the values of A: moment (∼ 2.87 µB/atom). The magnetization of Nd EXP atoms is decreased more rapidly with temperature than ˜ TC A(T )= A(T ) MC . (6) that of Fe atoms, which largely affects the temperature TC dependence of total magnetization. Thus, intrinsically n The inset of Fig. 6 shows the rescaled data and experi- depend on the fitting range largely. However, the impor- mental values (green bar) at RT [21, 22]. Although the tant point here is that nz always takes larger values than experimental values have some variation (6.6-12.5 pJ/m), nx regardless of the fitting range (we checked it). This the calculation results are well consistent with the lower relation implies the macroscopic exchange coupling in the experimental values at RT. z-direction is weaker than that in x-direction, not only With A and Ku which have been obtained, the DW for the coupling strength but also for thermal tolerance. width, δdw, is calculated from the following relation [15, As the reason for the anisotropy of A to crystal ori- 23] : entation, the crystal structure of Nd2Fe14B is naturally invoked. Nd2Fe14B has the layered structure of Fe- x(z) ∂ω(T ) Ax(z)(T ) layer and NdFe-layer (B has little effect on magnetic δdw (T )= π = π , (7) ∂θ θ= π s FA(T ) properties) along the z-axis as shown in Fig. 1 (a). In 2 ˜ θ Nd2Fe14B, exchange couplings (Jij in Eq. (1)) are mainly dθ ˜ ω(T )= Ax(z)(T ) . contributed by bonding between Fe and Fe atoms, JFe-Fe, 0 EK (T,θ) ˜ ˜ q Z and between Nd and Fe atoms, JNd-Fe (|JNd-Nd| is neg- ˜ To confirm the validity of our evaluationp procedure for ligibly small). Each bond of JFe-Fe has much larger am- A, in Fig. 7, we compared our results (circle) with those plitudes than J˜Nd-Fe (J˜Fe-Fe : −4.35-22.34 meV, J˜Nd-Fe : of the previous study (square) [26]. −0.16-3.55 meV). Therefore, it is anticipated that the In the previous study, δdw was evaluated directly from anisotropy of A comes from the inhomogeneous distribu- the snapshots of spin configurations using the same atom- tion of the exchange couplings and the atom positions istic Hamiltonian of the present study. Note that, we set in the crystal structure. To support this anticipation in 7

resents the grain boundary phase. The two models (a) 0.9 and (b) are the same if we do not take into account the anisotropy of A and the dipole-dipole interaction. The simulations are based on the finite-difference 0.8 method and the LLG equation [39, 40]:

x M d i |γ| eff α eff A = − M × H + M × (M × H ) , / 0.7 ˜ 2 i i M i i i z 2.0JNd-Fe dt 1+ α | i|

A   1.5J˜Nd-Fe (9) 0.6 ˜ 1.0JNd-Fe (default) where Mi is the magnetization vector of ith cell, γ is the

0.5J˜Nd-Fe gyromagnetic ratio constant, and α is Gilbert damping constant. Both models (a) and (b) are discretized with a 0.5 0 200 400 600 800 1000 cubic cell of (1.0 nm)3 and we set |γ| =2.21×105 m/A·sec Temp. (K) (the value of free electron) and α = 1 (coercivity does not Heff depend on α). Effective magnetic field on ith cell, i , is defined as the derivative of the micromagnetic energy, cont M FIG. 8. The anisotropy ratio of the exchange stiffness, Ei (obtained from Eq.(2)), with respect to i [47]: Az/Ax, as a function of temperature for the four models with 1 ∂Econt different exchange coupling between Nd and Fe, J˜Nd-Fe. Heff i e i = − + Hext u µ0 ∂Mi 2A m − m 2Ki = ij j i + 1 (m · e )e + H e , detail, we examine the relation between the anisotropy |M | d2 |M | i u u ext u ˜ j∈n.n. i ij i and the strength of JNd-Fe. X Figure 8 shows the ratio of Az to Ax for four models (10) with different values of J˜Nd-Fe. Beside the default model where, µ0 is the magnetic permeability of the vacuum, (1.0J˜Nd-Fe, the ratio is calculated from Fig. 6), we also Hext is an external magnetic field, j represents six nearest calculate other three cases with all the bonds J˜Nd-Fe re- neighbor cells of the ith cell, mi = Mi/|Mi|, eu is the duced by half (0.5J˜Nd-Fe), increased by half (1.5J˜Nd-Fe), unit vector of the easy axis, dij is the distance between and doubled (2.0J˜Nd-Fe). It is clearly found that A gets the centers of ith and jth cells (i.e. dij = 1.0 nm), Aij close to isotropic (i.e., Az = Ax) as J˜Nd-Fe increase in i is the exchange stiffness costant, and K1 is the magnetic the whole temperature range. i i anisotropy constant (terms of K2 and K4 were omitted As another feature, the ratio slowly decreases with the for simplicity). temperature for all the cases, which indicates that the In the present study, we determine the input model temperature dependence of Ax and Az are different. This parameters in Eq.(10) from the MC results at 400K. The difference corresponds to the difference between nx and model parameters in the hard phases are set to |Mi| = nz in Eq. (8). As the temperature increases, the contri- 1.38 T, Ki =2.63MJ/m3, A = 12.21pJ/m for the pairs ˜ ˜ 1 ij bution of JNd-Fe to A becomes smaller than that of JFe-Fe of iz = jz, and Aij =9.10pJ/m for the pairs of iz 6= jz, because the spin moments of Nd atoms are more easily where iz(jz) is the position of the i(j)th cell in the z- broken by thermal fluctuations compared with those of axis. In the soft phases, the model parameters are set to Fe atoms. From the above analysis, we conclude that the i 3 |Mi| = 1.38 T, K1 = 0MJ/m , and Aij = 9.10pJ/m for reason of the anisotropy of A in Nd2Fe14B comes from all the pairs of (i, j). The difference of A in the direction ˜ ij the weakness of JNd-Fe and the layered structure of Nd for the hard phases reflects the anisotropy of A in the MC atoms. results. In addition, we assume Aij =9.10pJ/m for the bonds connecting the soft/hard interfaces. By applying the fourth-order Runge-Kutta algo- C. Effect of anisotropic exchange on coercivity rithm [48] to the LLG equation with the above- constructed models, we simulated the magnetization re- Let us consider the effects of the anisotropy of A on the versal dynamics and then evaluated the coercivity. In coercivity. In actual rare-earth magnets which are com- the simulation, we set the Runge-Kutta time step to posed of main rare-earth magnet phase and (magnetic or 0.1 ps, the magnetic field (Hext) is reduced by 25mT non-magnetic) grain boundary phase, magnetization re- at each field step, and the convergence condition under versal is considered to occur by nucleation near the inter- each field is when the average of magnetization torque, m Heff face and by the DW propagation. Thus, to study magne- | i × i |, is lower than 1.0 mOe. Also, to avoid the m Heff tization reversal in such a process, we carried out micro- case of i × i ∼ 0, at the beginning of each field step, magnetic simulations for the two-phase models composed a disturbance is added to the magnetization vector of ev- of the soft magnetic phase and the hard magnetic phase, ery cell as mi → (mi +v)/|mi +v|, where v is a random depicted in Fig. 9 (a) and Fig. 9 (b). The soft phase rep- vector of length 10−4. Using the simulation conditions, 8

of the whole system occurs by a nucleation which starts from nucleation at the surface of the soft phase with- out depinning at the interface of the soft and hard parts, while in the depinning type, the reversed magnetization in the soft phase is pinned at the interface until the mag- netic field reaches the threshold of the depinning. Figure 9 (d) shows the hysteresis loops (showing the only upper part in the figure) for the model (a) with four different sl. When the soft phase is thin (sl = 1, 3 nm), the magnetization of the hard phase is reversed at the same time as the nucleation at the soft phase (i.e., the nucleation type), whereas when the soft phase is thick (sl =5, 7 nm), coercivity is determined not from the nu- cleation but from the depinning (i.e., the depinning type). The change between the nucleation type and depinning type were also pointed out in the previous studies for a one-dimensional model in which a soft phase of finite width is sandwiched between hard phases [9–12] and a corner defect model [49]. In the one-dimentional model an analytical solution of the coercivity in the limit sl → ∞, which means the depinning type, was also proposed as follows [10, 11]:

S S A K1 H 1 − H H 2K1 A K1 Hdwp = H 2 , (11) |M | S M S 1+ A | | FIG. 9. Calculation models with open boundary conditions AH|M H| in which the soft magnetic phase (SP) is placed on (a) (001)  q  M S(H) S(H) S(H) surface and (b) (100) surface of the hard magnetic phase. (c) where | |, K1 , and A are the model param- Coercivity without dipole-dipole interaction evaluated using eters of Eq. (10) in soft (hard) phase. To apply Eq. (11) each model including the MC results at 400 K as a function to the three-dimensional models, as the value of AH, we of soft phase thickness, sl. Dashed lines denote the analyt- use the value of Aij in the direction perpendicular to the ical results of the depinning type coercivity calculated form soft/hard interface, and calculate Hdwp. Dashed lines in Eq. (11). (d) Hysteresis loops for the model (a) with four Fig. 9 (c) indicate Hdwp for the models, which seems to different sl. explain well the lower limit of depinning type coercivity for our three-dimensional models. Therefore, it is under- stood that the anisotropy of A has a large effect on the the magnetization reversal of the hard phase (not includ- coercivity of the depinning type compared with that of ing the soft phase) occurs at 4.80 T which is consistent the nucleation type. with the Stoner-Wohlfarth limit, 2K /M =4.789 T. 1 s Finally, we study the influence of the dipole-dipole in- In such the two-phase models, a magnetization rever- teraction on the coercivity in the two models. Magnetic sal is expected to start from the soft phase. Thus, we field due to the dipole-dipole interaction is incorporated also examine the influence of soft phase thickness, sl. In Heff in i as the following form: Fig. 9 (c), we plot sl dependence of the coercivity for the models (a) and (b). It is clearly seen that the coercivity Hdip r M i = K( ij ) j , (12) of the model (a) is weaker than that of the model (b) j regardless of sl. The relation of the between X the two models is also consistent with that of the magsni- where K(rij ) is the demagnetization tensor [40], rij is the tude of DW energy in each direction (see Fig. 5). Since distance vector between i and j cells. Here the strength of the models (a) and (b) are equivalent in the absence of the dipole-dipole interaction is determined automatically anisotropy of Aij in the hard phase, we can conclude according to the distance of the pair and the magnetiza- Hdip that the difference of the coercivityu is attributed to the tion vector of the cells. We calculate i of all the cells anisotropic A. in O(NclogNc) computational time (Nc is the total num- It is also seen that as sl increases, the coercivity de- ber of cells) by solving convolution integral using the fast creases and gradually approaches a certain value for each Fourier transform method [50]. In the present study, we model, and the difference between two models increases. set the models (a) and (b) in Fig. 9 are cubic regardless H S As we will see in the following two paragraphs, this be- of sl and M = M , and thus the models (a) and (b) havior is explained as a change in magnetization reversal have the same shape magnetic anisotropy. mechanism from nucleation type to depinning type. Here Upper figures of Fig. 10 show sl dependence of coerciv- we define the nucleation type as a magnetization reversal ity for the two models (a) and (b) with dipole-dipole in- 9

the higher magnetic field where the DW is depinned at the interface (depinning). In contrast, in the model (a) they are not clearly distinguished. That is, in the model (a), the magnetization reversal mechanism becomes to approach the nucleation type from the depinning type by the dipole-dipole interaction. The dependence of the coercivity on the arrangement of the soft phase were sim- ilarly discussed in the most recent study (not including the anisotropy of A) [51]. It is difficult to clarify the effect of anisotropic A on the coercivity under the dipole-dipole interaction by a simple comparison the models (a) and (b). Thus, we ex- changed the values of Ax,y and Az in the hard phase. Namely, we set the input parameters of the hard phase as Aij =9.10pJ/m for iz = jz, and Aij = 12.21pJ/m for iz 6= jz. The values of coercivity under these conditions are plotted by the blue circles and the blue lines (Hdwp) in the upper figures of Fig. 10. The anisotropy of A has a similar effect on the coercivity as the case without dipole- dipole interaction. However, in the model (a), the differ- ence in coercivity is relatively small. The anisotropic A strongly affects the coercivity of the depinning type com- pared with the nucleation type. On the other hand, The magnetization reversal in the model (a) is the nucleation type rather than the depinning type. Therefore, we may conclude the dipole-dipole interaction in the model (a) suppresses the effect of the anisotropy of A.

FIG. 10. (upper figures) Coercivity with dipole-dipole inter- IV. CONCLUSION action for the models (a) and (b) in Fig. 9, as a function of soft phase thickness, sl. Dashed lines represent Hdwp calcu- lated from Eq. (11). (lower figures) Hysteresis loops for the Regarding the exchange stiffness constant, A, of Nd Fe B, we examined the temperature and orienta- two models in the cases of Ax > Az (red circles in the upper 2 14 figures) with four different sl. tion dependences using the Monte Carlo simulations with the atomistic spin model constructed from the ab-initio calculation. We also conducted the coercivity calcula- tions based on the micromagnetics (LLG) simulations us- teraction. The red circles and the red dotted lines (Hdwp) ing the continuum model with the parameters obtained represent the values of the coercivity which are calculated by the atomic scale MC results at finite temperatures. i using the same input parameters (|Mi|, K1, and Aij ) as In this way, we confirmed that the lattice structure in those in Fig. 9 (c). Here, the difference of coercivity in the atomic scale affects the coercivity as macroscopic the models (a) and (b) in Fig. 10 is mainly attributed physics. We found that A(T ) depends on the orienta- to the dipole-dipole interaction, and the anisotropic A tion of the crystal with respect to not only the ampli- is a secondary effect. Because dipole-dipole interaction tude but also the exponent nx(z) in the scaling behav- prefers to construct the DW along z-axis (type I), in the nx(z) ior: Ax(z)(T ) ∝ M(T ) ; namely Ax(T ) > Az(T ) model (b), the coercivity decreases compared with that and nx < nz. It is quantitatively confirmed that the in Fig. 9 (c). Conversely in the model (a), the coercivity anisotropic properties of A come from the weak exchange increases in the region of sl ≥ 3nm. The dipole-dipole couplings between Nd and Fe atoms and the layered interaction inhibits to construct the DW (nucleation) in structure of Nd atoms. Moreover, we also found that the xy-plane. For this reason, the coercivity with dipole- anisotropic A(T ) affects the coercivity of the depinning dipole interaction depends on the arrangement of the soft mechanism. phase, which works contrary to the effect of anisotropic We focused on only Nd2Fe14B magnet in the present exchange stiffness, Ax > Az. paper. However, the essence of anisotropic A comes These behaviors are confirmed from the hysteresis from the weak exchange coupling between rare-earth loops in lower figures in Fig. 10. In the model (b), mag- atoms and transition metals and the layered structure netization reversal is clearly separated into the two parts, of rare-earth atoms. Thus, the features discussion for i.e., the small jump at the lower magnetic field where only Nd2Fe14B are probably applicable to other rare-earth the soft phase is reversed (nucleation), and the jump at magnets. In fact, it was pointed out by ab-initio calcu- 10 lations that A have strong anisotropy for YCo5 [24] and the accuracy multiscale analysis needs further develop- Sm(Fe,Co)12 [25]. ment of connecting scheme from atomistic spin model to Let us consider the coercivity from the viewpoint of macrospin model would be necessary. the exchange spring magnet [52, 53] which is composed As another point noted is the range of exchange in- of hard and soft phase and expected to realize the highest teraction. In the atomistic spin model, Eq. (1), of the performance magnet. Because realization of high perfor- present study, we omitted the long-range contribution of mance requires a large coercivity and a large thickness J˜ij for simplicity and reduction of calculation cost. How- of soft phase, the model (a) with Ax < Az (Fig. 10 (a) ever, the recent study[62] reported that RKKY-type ex- blue line) is the most suitable conditions in our model- change coupling significantly effects on the DW width, ings. Most strong permanent magnets, Nd2Fe14B, YCo5 and pointed out the importance of the long-range contri- and also L10-type magnet (CoPt, FePd, FePt) cannot bution. We also found that, by incorporating long-range reproduce the same condition because Ax > Az [24], exchange couplings up to 10.6 A,˚ the difference between whereas Sm(Fe,Co)12 would do because the anisotropy type I and type II of Edw at 400K reduces from 11.3 % as Ax < Az [25]. Therefore, Sm(Fe,Co)12 and other (in Fig. 5) to 4.4 %. However, to make study the effect R(Fe,Co)12-type compounds (R is a rare-earth element) clearly, we need precise information of the interaction at may have higher potential to realize strong performance the long distance, and we postpone to study this problem exchange spring magnet rather than the other magnets. later. Finally, we point out another source of the anisotropy. Although there are still problems that must be con- In a recent experiment, it has been observed that the cerned, we believe that the present paper will be helpful grain boundary phase takes different crystal structures to elucidate coercivity mechanism in rare-earth perma- and chemical compositions depending on the orientation nent magnets. with the Nd2Fe14B main phase, i.e., the Nd-rich crys- talline paramagnetic phase form on the xy-plane of the main phase, whereas the Fe-rich amorphous ferromag- ACKNOWLEDGMENTS netic phase in the plane parallel to the z-axis [54]. In the present paper, we have studied the effect of anisotropy We acknowledge collaboration and fruitful discussions of A on coercivity and of the orientation of the interface with Taro Fukazawa, Taichi Hinokihara, Shotaro Doi, with the soft phase changes by using the same interac- Munehisa Matsumoto, Hisazumi Akai, and Satoshi Hiro- tion for the interface. However, if the chemical structure sawa. This work was partly supported by Elements Strat- is different, the exchange interaction would be a differ- egy Initiative Center for Magnetic Materials (ESICMM) ence due to another source of the anisotropy, which is under the auspices of MEXT; by MEXT as a social and studied with information of the structure in the future. scientific priority issue (Creation of New Functional De- In the continuum model, Eq. (2), we used the values vices and High-Performance Materials to Support Next- of Ku and A obtained by the MC simulations which Generation Industries; CDMSI) to be tackled by using are the values of the bulk system. There, changes in a post-K computer. The computation was performed on atomic scale of magnetic anisotropy [55–57] and exchange Numerical Materials Simulator at NIMS; the facilities of coupling [58–61] near the interface or surface were not the Supercomputer Center, the Institute for Solid State taken into consideration. The influences of interface Physics, the University of Tokyo; the supercomputer of and surface are important for the coercivity [6]. Thus, ACCMS, Kyoto University.

[1] K. H. J. Buschow, Rep. Prog. Phys. 40, 1179 (1977). [10] A. Sakuma, S. Tanigawa, and M. Tokunaga, [2] T. Miyake and H. Akai, J. Magn. Magn. Mater. 84, 52 (1990). J. Phys. Soc. Jpn. 87, 041009 (2018). [11] H. Kronmller and D. Goll, [3] M. Sagawa, S. Fujimura, H. Yamamoto, Y. Matsuura, Physica B: Condensed Matter 319, 122 (2002). and K. Hiraga, IEEE Trans. Magn. 20, 1584 (1984). [12] S. Mohakud, S. Andraus, M. Nishino, A. Sakuma, and [4] J. J. Croat, J. F. Herbst, R. W. Lee, and F. E. Pinkerton, S. Miyashita, Phys. Rev. B 94, 054430 (2016). J. Appl. Phys. 55, 2078 (1984). [13] S. Hirosawa, Y. Matsuura, H. Yamamoto, [5] J. F. Herbst, J. J. Croat, F. E. Pinkerton, and W. B. S. Fujimura, M. Sagawa, and H. Yamauchi, Yelon, Phys. Rev. B 29, 4176 (1984). Jpn. J. Appl. Phys. 24, L803 (1985). [6] S. Hirosawa, M. Nishino, and S. Miyashita, [14] O. Yamada, H. Tokuhara, F. Ono, Adv. Nat. Sci: Nanosci. Nanotechnol. 8, 013002 (2017). M. Sagawa, and Y. Matsuura, [7] D. Givord, M. Rossignol, and V. M. T. S. Barthem, J. Magn. Magn. Mater. 5457, Part 1, 585 (1986). J. Magn. Magn. Mater. 258-259, 1 (2003). [15] K.-D. Durst and H. Kronm¨uller, [8] J. Fischbacher, A. Kovacs, M. Gusenbauer, J. Magn. Magn. Mater. 59, 86 (1986). H. Oezelt, L. Exl, S. Bance, and T. Schrefl, [16] R. Skomski, J. Appl. Phys. 83, 6724 (1998). J. Phys. D: Appl. Phys. (2018), DOI: 10.1088/1361-6463/aab[17]7d1. R. Sasaki, D. Miura, and A. Sakuma, [9] D. I. Paul, J. Appl. Phys. 53, 1649 (1982). Appl. Phys. Express 8, 043004 (2015). 11

[18] Y. Toga, M. Matsumoto, S. Miyashita, Handbook of Magnetism and Advanced Magnetic Materials, H. Akai, S. Doi, T. Miyake, and A. Sakuma, Vol. 2 (Wiley, New York, 2007) pp. 742–764. Phys. Rev. B 94, 174433 & 219901 (2016). [40] Y. Nakatani, Y. Uesaka, and N. Hayashi, [19] H. M. Mayer, M. Steiner, N. Ster, H. Wein- Jpn. J. Appl. Phys. 28, 2485 (1989). furter, K. Kakurai, B. Dorner, P. A. Lindgrd, [41] K. Momma and F. Izumi, K. N. Clausen, S. Hock, and W. Rodewald, J. Appl. Crystallogr. 44, 1272 (2011). J. Magn. Magn. Mater. 97, 210 (1991). [42] P. Asselin, R. F. L. Evans, J. Barker, R. W. Chantrell, [20] H. M. Mayer, M. Steiner, N. Ster, H. We- R. Yanes, O. Chubykalo-Fesenko, D. Hinzke, and infurter, B. Dorner, P. A. Lindgrd, K. N. U. Nowak, Phys. Rev. B 82, 054415 (2010). Clausen, S. Hock, and R. Verhoef, [43] J. Smit and H. Wijn, Ferrites (Philips Technical Library, J. Magn. Magn. Mater. 104-107, 1295 (1992). 1959). [21] J.-P. Bick, K. Suzuki, E. P. Gilbert, E. M. Forgan, [44] L. N. Bulaevkiˇi and V. L. Ginzburg, R. Schweins, P. Lindner, C. Kbel, and A. Michels, Sov. Phys. JETP 18, 530 (1964). Appl. Phys. Lett. 103, 122402 (2013). [45] N. Kazantseva, R. Wieser, and U. Nowak, [22] K. Ono, N. Inami, K. Saito, Y. Takeichi, Phys. Rev. Lett. 94, 037206 (2005). M. Yano, T. Shoji, A. Manabe, A. Kato, [46] H. Kronmller (John Wiley & Sons, Ltd, 2007). Y. Kaneko, D. Kawana, T. Yokoo, and S. Itoh, [47] H. Tsukahara, K. Iwano, C. Mitsumata, T. Ishikawa, and J. Appl. Phys. 115, 17A714 (2014). K. Ono, AIP Advances 8, 056226 (2018). [23] S. Chikazumi, Physics of Ferromagnetism, International [48] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and Series of Monographs on Physics (Oxford University B. P. Flannery, Numerical Recipes 3rd Edition: The Art Press, Oxford, New York, 1997). of Scientific Computing, 3rd ed. (Cambridge University [24] K. D. Belashchenko, J. Magn. Magn. Mater. 270, 413 (2004). Press, Cambridge, UK ; New York, 2007). [25] T. Fukazawa, H. Akai, Y. Harashima, and T. Miyake, [49] S. Bance, H. Oezelt, T. Schrefl, G. Ciuta, N. M. arXiv:1803.03165 (2018). Dempsey, D. Givord, M. Winklhofer, G. Hrkac, [26] M. Nishino, Y. Toga, S. Miyashita, H. Akai, A. Sakuma, G. Zimanyi, O. Gutfleisch, T. G. Woodcock, and S. Hirosawa, Phys. Rev. B 95, 094429 (2017). T. Shoji, M. Yano, A. Kato, and A. Manabe, [27] T. Hinokihara, M. Nishino, Y. Toga, and S. Miyashita, Appl. Phys. Lett. 104, 182408 (2014). Phys. Rev. B 97, 104427 (2018). [50] N. Hayashi, K. Saito, and Y. Nakatani, [28] D. Hinzke, U. Nowak, R. W. Chantrell, and O. N. Jpn. J. Appl. Phys. 35, 6065 (1996). Mryasov, Appl. Phys. Lett. 90, 082507 (2007). [51] W. Li, L. Z. Zhao, Q. Zhou, X. C. Zhong, and Z. W. [29] D. Hinzke, N. Kazantseva, U. Nowak, O. N. Liu, Comput. Mater. Sci. 148, 38 (2018). Mryasov, P. Asselin, and R. W. Chantrell, [52] E. F. Kneller and R. Hawig, Phys. Rev. B 77, 094407 (2008). IEEE Trans. Magn. 27, 3588 (1991). [30] U. Atxitia, D. Hinzke, O. Chubykalo-Fesenko, U. Nowak, [53] R. Skomski and J. M. D. Coey, H. Kachkachi, O. N. Mryasov, R. F. Evans, and R. W. Phys. Rev. B 48, 15812 (1993). Chantrell, Phys. Rev. B 82, 134440 (2010). [54] T. T. Sasaki, T. Ohkubo, and K. Hono, [31] R. Moreno, R. F. L. Evans, S. Khmelevskyi, M. C. Acta Mater. 115, 269 (2016). Mu˜noz, R. W. Chantrell, and O. Chubykalo-Fesenko, [55] H. Moriya, H. Tsuchiura, and A. Sakuma, Phys. Rev. B 94, 104433 (2016). J. Appl. Phys. 105, 07A740 (2009). [32] S. Miyashita, M. Nishino, Y. Toga, T. Hinoki- [56] Y. Toga, T. Suzuki, and A. Sakuma, hara, T. Miyake, S. Hirosawa, and A. Sakuma, J. Appl. Phys. 117, 223905 (2015). Scripta Mater. 154, 259 (2018). [57] Y. Tatetsu, S. Tsuneyuki, and Y. Gohda, [33] S. C. Westmoreland, R. F. L. Evans, G. Hrkac, T. Schrefl, Phys. Rev. Applied 6, 064029 (2016). G. T. Zimanyi, M. Winklhofer, N. Sakuma, M. Yano, [58] R. F. Sabiryanov and S. S. Jaswal, A. Kato, T. Shoji, A. Manabe, M. Ito, and R. W. Phys. Rev. B 58, 12071 (1998). Chantrell, Scripta Mater. 148, 56 (2018). [59] Y. Toga, H. Moriya, H. Tsuchiura, and A. Sakuma, [34] K. W. H. Stevens, Proc. Phys. Soc. A 65, 209 (1952). J. Phys.: Conf. Ser. 266, 012046 (2011). [35] M. Yamada, H. Kato, H. Yamamoto, and Y. Nakagawa, [60] D. Ogawa, K. Koike, S. Mizukami, T. Miyazaki, Phys. Rev. B 38, 620 (1988). M. Oogane, Y. Ando, and H. Kato, [36] A. I. Liechtenstein, M. I. Katsnelson, Appl. Phys. Lett. 107, 102406 (2015). V. P. Antropov, and V. A. Gubanov, [61] N. Umetsu, A. Sakuma, and Y. Toga, J. Magn. Magn. Mater. 67, 65 (1987). Phys. Rev. B 93, 014408 (2016). [37] http://kkr.issp.u-tokyo.ac.jp/. [62] A. F. da Silva Jnior, M. F. de Campos, and A. S. Mar- [38] Y. Miura, H. Tsuchiura, and T. Yoshioka, tins, J. Magn. Magn. Mater. 442, 236 (2017). J. Appl. Phys. 115, 17A765 (2014). [39] J. E. Miltat and M. J. Donahue, in