PHYSICAL REVIEW B 101, 094421 (2020)

Magnetic field threshold for nucleation and depinning of domain walls in the neodymiumpermanent magnet Nd2Fe14B

Ismail Enes Uysal ,1 Masamichi Nishino ,2,1,* and Seiji Miyashita3,4,5,1

1Elements Strategy Initiative Center for Magnetic Materials, National Institute for Materials Science,1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan

2Research Center for Advanced Measurement and Characterization, National Institute for Materials Science,1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan

3Department of Physics, Graduate School of Science, the University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan4The Physical Society of Japan, 2-31-22 Yushima, Tokyo 113-0033, Japan

5Institute for Solid State Physics, the University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8581, Japan

(Received 9 November 2019; revised manuscript received 29 January 2020; accepted 18 February 2020;published 17 March 2020)

The Nd magnet Nd2Fe14B is an indispensable permanent magnet for high-technology commercial products.We study magnetization reversal in hard-soft-hard magnet systems of the atomistic model of the Nd magnet atzero and room temperatures. We present the nucleation and pinning fields as a function of the strength of theanisotropy energy of the soft magnet, estimated by using the stochastic Landau-Lifshitz-Gilbert equation. Inthe Nd magnet, reflecting the lattice structure, the properties of domain walls (DWs) depend on their movingdirections. Bloch and Néel DWs move along the a (or b) and c axes, respectively. We investigate the differencein the nucleation and pinning fields between the Bloch and Néel DWs. We also analyze the dependence ofthe threshold fields on the thickness of the soft magnet. We find that the thermal fluctuation effect affects thethreshold fields in the Nd magnet model more significantly than those in the simple anisotropic Heisenberg model[Mohakud et al., Phys. Rev. B 94, 054430 (2016)]. We also find that the strength of the anisotropy energies ofthe soft magnet phase is not so important for the pinning field, while it is essential for the nucleation field, etc.We discuss the detailed features of the dependencies and their microscopic mechanisms.

DOI: 10.1103/PhysRevB.101.094421

I. INTRODUCTION

The Nd magnet Nd2Fe14B [1–13] is an important high-performance permanent magnet. Because of its high coer-cive force, it is widely used for electric motors, electronicdevices, etc. [14]. Trials toward higher coercivity at highertemperatures have been actively performed [15,16], but themechanism of the coercive force has not been well understood[17]. It has been clarified that the coercive force essentiallydepends on the structure of the grains and grain boundaries,and the nucleation and depinning mechanisms in the structureplay an important role in the coercivity [14,17,18].

To realize stronger coercive forces at higher temperatures,it is necessary to study the effect of the structures and proper-ties of the grains and grain boundaries on the coercive force.For this purpose, a prototype hard-soft-hard magnet model,in which hard magnets contact with a middle soft magnet,has been intensively studied [19–23]. This model catchesthe essence of nucleation and depinning in inhomogeneoussystems, and has been frequently used in analyses of thephenomena in various experimental and theoretical studies ofmagnetic materials [24,25] including Giant magnetoresistance(GMR) sensors [26].

*Corresponding author: nishino.masamichi@nims.go.jp

Sakuma et al. investigated the threshold fields for nucle-ation and depinning in a hard-soft-hard magnet continuummodel at zero temperature [20,21]. Solving a one-dimensionalnonlinear equation for the model with the exchange stiffnessconstant and magnetocrystalline anisotropy energy, they pre-sented a phase diagram of the threshold fields as a functionof the ratios between the stiffness constants and anisotropyconstants of the soft and hard magnets. However, thermal fluc-tuation effects, which are also very essential for the coercivity,were not studied.

Recently, Mohakud et al. [22] studied the temperaturedependence of the corresponding phase diagram for thehard-soft-hard magnet model in the simple cubic latticeof the Heisenberg model with the anisotropy term (calledan anisotropic Heisenberg model) by solving the stochas-tic Landau-Lifshitz-Gilbert (SLLG) equation [27,28]. Theyshowed various parameter dependencies at different temper-atures, and the threshold fields are significantly affected bythe thermal effect.

In the present paper, we develop this work [22] and investi-gate the threshold fields at zero and room temperatures in thehard-soft-hard magnet model of the atomistic model of the Ndmagnet [29–31], in which the microscopic parameters weretaken mainly from first-principles calculations (see Sec. II A).

So far, magnetic properties of permanent magnets havebeen intensively performed in micromagnetics. However, in

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UYSAL, NISHINO, AND MIYASHITA PHYSICAL REVIEW B 101, 094421 (2020)

FIG. 1. Unit cell of Nd2Fe14B. Neodymium, Iron, and Boronatoms are denoted by yellow, blue, and green spheres, respectively.The lattice constants [2] for the a, b, and c axes are da = db = 8.80 Å,and dc = 12.19 Å, respectively.

such coarse-grained modeling, the details of crystal structuresand magnetic parameters are difficult to introduce and thetemperature and thermal fluctuation effect cannot be treatedexactly [28]. For the understanding of the microscopic mech-anisms of magnetic properties such as coercivity, atomisticmodel studies are indispensable. Very recently, a generationof the permanent magnet study, especially for the Nd mag-net, based on the atomistic modeling was performed. Thismodeling enables us to overcome the weak points of thecoarse-grained modeling.

Using the atomistic model, various important features offinite temperature properties on the magnetization [29–32],domain walls (DWs) [30,32,33], dipolar interaction effect[31], ferromagnetic resonance [34], etc. in the Nd magnet havebeen elucidated. Similar atomistic models of the Nd magnethave been actively investigated [35–37]. Here we study thedependence of the threshold fields on the ratio of the magneticanisotropy constants between the hard and soft magnet phases,and also on the width of the soft magnet phase. In the Nd

magnet, Nd layers are located perpendicular to the c axis (easyaxis) as in Fig. 1, and the nature of the magnetic DW dependson its moving direction. There are two types of DWs, i.e., theBloch DW moving along the a or b axis and the Néel DWmoving along the c axis [30] (Fig. 2). We also analyze thedependence of the threshold fields on the types of DWs.

The paper is organized as follows. In Sec. II, the model andthe method are presented. In Sec. III, the threshold fields fornucleation and depinning are studied at zero temperature. InSec. IV, the threshold fields for nucleation and depinning areinvestigated at T = 300 K, and the comparison between thezero temperature and 300 K is examined. The dependence ofthe threshold fields on the soft magnet width is also discussed.Section V is devoted to the summary.

II. MODEL AND METHOD

A. Atomistic model for Nd2Fe14B

We adopt the following atomistic Hamiltonian for the Ndmagnet Nd2Fe14B:

H = −∑

i< j

2Ji jsi · s j −Fe∑

i

Di(sz

i

)2

+Nd∑

i

∑

l,m

�l,iAml,i〈rl〉iO

ml,i −

∑

i

hexti · Si, (1)

where Ji j is the exchange interaction between the ith and jthsites, Di is the magnetic anisotropy constant for Fe atoms, thethird term is the magnetic anisotropy energy of Nd atoms, andhext

i is the external magnetic field applied to the ith site. For Feand B atoms, si denotes the magnetic moment at the ith site,but for Nd atoms, it is the moment of the valence (5d and 6s)electrons which is coupled antiparallel to the moment of the4 f electrons J i. Thus, the total moment at the ith site for Ndatoms is Si = si + J i. Here Ji = gTJμB, in which gT = 8/11is the Landé g-factor and J = 9/2 is the magnitude of the totalangular momentum. For Fe and B atoms, we define Si = si.

(a) (b)

L1 L2 L3

ab

c

c

ab

L1 L2 L3

(c) (d)

FIG. 2. Systems of two bulk hard magnets (regions I and III) and a boundary soft magnet (region II). (a) System A, in which a domain wallmoves along the a axis (Bloch wall) and (b) system B, in which a domain wall moves along the c axis (Néel wall). The lower crystal structureis a view from the b (a) axis for system A (B). The Nd layers are located perpendicular to the c axis. (see also Fig. 1). (c) Bloch domain walland (d) Néel domain wall.

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In the third term, the summation for l runs l = 2, 4, 6,where �l,i, Am

l,i, 〈rl〉i, and Oml,i are the Stevens factor, the

coefficient of the spherical harmonics of the crystalline elec-tric field, the average of rl over the radial wave function,and Stevens operator, respectively. The diagonal operators(m = 0) give the dominant contribution and the third term isexpressed approximately as

Nd∑

i

∑

l,m

�l,iAml,i〈rl〉iO

ml,i

�Nd∑

i

DNd1

(Jz

i

)2 + DNd2

(Jz

i

)4 + DNd3

(Jz

i

)6 + const. (2)

We use the same parameter values for the atomistic modelas in our previous studies [29–31]. The exchange interactionswithin the range of r = 3.52 Å [29,30] were obtained by afirst-principles calculation with the Korringa-Kohn-Rostoker(KKR) Green’s function method [38] and the anisotropy ener-gies of Fe atoms Di (for six kinds of sites), estimated in a first-principles study [39], were used. For Nd atoms, the estimationof Am

l by first-principles calculations has not been established,and experimentally determined Am

l [40] were applied with 〈rl〉estimated in Ref. [41].

Applying those parameter values, we obtained the spin-reorientation temperature Tr = 150 K, which is very closeto experimentally estimated ones Tr = 133−150 K [7,8,10–13]. The critical temperature Tc ∼ 850 K [29,30] is a little bitoverestimated to the experimental values [4,7] Tc ∼ 600 K,but this difference does not essentially affect the discussionand conclusion in the present paper.

B. Systems

The unit cell for Nd2Fe14B, which contains 68 atoms, isshown in Fig. 1. We consider two kinds of systems (systemsA and B) with open boundary conditions, consisting of twobulk hard magnets and a boundary soft magnet (Fig. 2). Insystem A, the hard, soft, and hard magnetic regions are locatedalong the a axis, while in system B, they are located alongthe c axis. Under a reversed field hz parallel to the c axis, aBloch DW moves along the a axis in system A, while a NéelDW moves along the c axis in system B. The system size isLa × Lb × Lc unit cells, where La, Lb, and Lc are the numberof unit cells in the a, b, and c axes, respectively. For systemA (B), La(Lc) = L1 + L2 + L3, in which L1, L2, and L3 arethe numbers of unit cells for the hard, soft, and hard magnetregions along the a (c) axis, respectively.

The structure of the soft magnet (grain boundary) in realNd magnets is complicated and depends on the surroundingsituation. Although challenges to the estimation of thosestructures by first-principles calculations have begun [42], thedetails of the structures are not well understood. Thus forsimplicity, in the present paper we adopt the same structureof the unit cell in the soft magnet as in the hard magnet. Inthe previous studies [20], the following two parameters aredefined to characterize the hard-soft-hard magnet:

F = A2M2

A1M1(3)

and

E = A2K2

A1K1. (4)

Here A1 and K1 are the exchange stiffness and magneticanisotropy constants, respectively, for the bulk hard magnet,while A2 and K2 are for the soft magnet. M1 and M2 aremagnetizations in the hard and soft magnets, respectively.

Following the previous study [22], we set M1 = M2, i.e.,the same magnetization in the soft and hard magnets. We alsodefine F and E for the atomistic model as

F = Jsofti j

Jhardi j

(5)

and

E = FDsoft

i

Dhardi

. (6)

It should be noted that in the soft magnet region, for Fe atomsDi at site i is rescaled by E/F , for Nd atoms at site i DNd

1 , DNd2 ,

and DNd3 are rescaled by E/F (for B atoms, Di = 0), and all

the exchange interactions Jsofti j are rescaled by F . We give Jsoft

if the exchange interaction belongs to the soft magnet region,and Jhard for the exchange coupling strength at the interfaceand in the hard magnet region. If Jsoft is given for the interfaceexchange, the difference in the nucleation and pinning fieldsis negligible small.

In a previous paper [30], we estimated the DW widths ofthe Bloch and Néel types at 300 K to be 6.38 nm and 5.47 nm,respectively. Throughout this study, we set L1 = L3 = 12 forthe hard magnet regions, and this length is large enoughcompared to the DW widths. For systems A (B), Lb = Lc = 5(La = Lb = 5) are set. We treat L2 as a variable which takesL2 = 1 − 14 for the soft magnet. Here we focus on the case ofF = 0.5.

C. Dynamical method

The time evolution of the ith spin in the system is given bythe SLLG equation[27,28]:

d

dtSi = − γ

1 + α2i

Si × heffi − αiγ(

1 + α2i

)Si

Si × [Si × heff

i

].

(7)

The parameter γ denotes the electron gyromagnetic ratio andαi is the damping parameter. The effective field heff

i on the ithspin is given by

heffi = −∂H

∂Si+ ξi(t ), (8)

where the thermal effect is included by adding a white Gaus-sian noise field, ξi(t ) = (ξ x

i , ξyi , ξ z

i ), into heffi . The noise field

satisfies the following properties:⟨ξ

μi (t )

⟩ = 0,⟨ξ

μi (t )ξν

j (s)⟩ = 2Diδi jδμνδ(t − s). (9)

Here, Di is the amplitude of the noise. When the relation

Di = αikBT

γ Si(10)

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UYSAL, NISHINO, AND MIYASHITA PHYSICAL REVIEW B 101, 094421 (2020)

is satisfied, the system relaxes to a steady state (equilibrium)in the canonical distribution of the temperature T . Simulationsare carried out by solving Eq. (7) numerically. We treat thisstochastic differential equation in Stratonovich interpretation.According to Appendix B in Ref. [28], we use a kind ofmiddle-point method equivalent to the Heun method for thenumerical integration. Here αi is set to 0.1 and the time stepfor the SLLG equation t = 0.1 fs is used. We measure thetime dependence of the magnetization defined in each unitcell,

mz =∑

i Si,z

Nunit, (11)

where Nunit is the number of the atoms in a unit cell.

D. Magnetic fields for nucleation and depinning

We study the nucleation and pinning fields under thereversed field hext

i = hzez, parallel to the c axis. Following theprevious work [22], the final configurations are classified byspecifying the signs of the magnetizations in the three regions(mI, mII, mIII).

For the initial condition of (+ + +), there are three possi-ble final configurations, i.e., (+ + +): no nucleation occurs,(+ − +): nucleation occurs in the soft magnet region II butthe reversed magnetization does not grow to the hard magnetregions (I or III), and (− − −): the reversed magnetizationgrows to the regions I and III. We define hNCII(T ) as theboundary between the fields for which the final configurationsare (+ + +) and (+ − +). We also define hNC(T ) as theboundary between the fields for which the final configura-tions are (+ − +) and (− − −). In some parameter regions,(+ + +) directly changes to (− − −), in which hNCII(T ) =hNC(T ) is realized. Examples of the generation of (+ − +)and (− − −) are given in Figs. 3(a) and 3(b), respectively.The magnetization mz [μB] of the unit cell along the a axis(the horizontal axis), located in the middle of the bc plane, isplotted with the time dependence (the vertical axis) for systemA with L1 = L2 = L3 = 12 (La = 36).

On the other hand, for the initial condition (+ + −), thereare two possible final configurations, i.e., (+ − −): the DWpropagation stops at the interface between the hard and softmagnet regions, and (− − −): the DW propagates to the hardmagnet region and full magnetization reversal occurs. Thepinning field, hDWP(T ), is defined as the boundary betweenthe fields for which the final configurations are (+ − −) and(− − −).

In the following sections, simulations are performed withNt = 1 × 106 time steps (t = 0.1 ns) for zero-temperatureproperties. At zero temperature, there is no thermal fluctua-tion and the threshold field corresponds to that, vanishes thepotential barrier, and the simulation time is short. At finitetemperatures, a stochastic process to jump over the energybarrier occurs by the thermal fluctuation, and the relaxationoccurs stochastically. We measure the magnetization reversalwithin Nt = 5 × 106 time steps (t = 0.5 ns). Compared to theexperimental observation time for the coercive force, i.e., 1 s,the simulation time is very short. However, the dependence ofthe relaxation time on the external field around the thresholdvalue is very sharp, i.e., the relaxation time exponentially

0

0.02

0.04

0.06

0.08

0.1

0 4 8 12 16 20 24 28 32 36

time

[ns]

x-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5

0

0.02

0.04

0.06

0.08

0.1(a)

(b)

0 4 8 12 16 20 24 28 32 36

time

[ns]

x-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5

FIG. 3. Time evolution of mz along the a axis for E = 0.1 when(a) h = 11.5 T and (b) h = 11.7 T at T = 0 K. x is the position ofthe unit cell along the a axis, located in the middle of the bc plane.The color bars denote the magnitude of the Bohr magneton (μB).

increases [22]. Thus we expect that the estimated thresholdfields can give an approximate estimation for the coerciveforce.

III. ZERO TEMPERATURE PROPERTIES

A. Nucleation

We show in Fig. 4(a) threshold fields at T = 0 K fordifferent final configurations starting from the initial condition(+ + +) for F = 0.5 for systems A and B, in which Bloch andNéel DWs appear, respectively. In this section, we set L2 =12. As mentioned above, for the field for nucleation, there aretwo kinds of thresholds, hNCII and hNC. In Fig. 4(a), the borderhNCII between (+ + +) and (+ − +) is given by circles, whilethe border hNC between (+ − +) and (− − −) is given bytriangles. Figures 3(a) and 3(b) present the time evolutions fornucleation under the field h = 11.5 T, which is between hNCII

and hNC, and the field h = 11.7 T over hNC, respectively. Itshould be noted that when E � 0.2, the final configuration iseither (+ + +) or (− − −). Namely, hNCII = hNC. While forE < 0.2, hNCII �= hNC and the region of the final configuration(+ − +) appears. This result is qualitatively similar to that

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0

5

10

15

20

25

0.1 0.2 0.3 0.4 0.5

h [ T

]

E

Neel,+++/+-+Neel,+-+/---

Bloch,+++/+-+Bloch,+-+/---

0

5

10

15

20

25

0.1 0.2 0.3 0.4 0.5

h [ T

]

E

Neel,+--/---Bloch,+--/---

(a) (b)

FIG. 4. Threshold fields at T = 0 K for different final configurations starting from the initial condition (+,+,+) for F = 0.5 for systemsA and B, in which Bloch and Néel domain walls, respectively, appear. In the left figure (a) for nucleation, the border hNCII between (+ + +)and (+ − +) is given by circles, while the border hNC between (+ − +) and (− − −) is given by triangles. In the right figure (b) for domainwall pinning, the border hDWP between (+ − −) and (− − −) is given by triangles.

of the continuum model [20] and the anisotropic Heisenbergmodel [22]. We find that hNCII and hNC for the Bloch DWare larger than those for the Néel DW. It is considered thatthe stronger effective exchange interaction along the a (or b)axis than that along the c axis [30,33] prevents the generationof DWs.

B. Depinning

In Fig. 4(b), threshold fields at T = 0 for final configura-tions starting from the initial condition (+ + −) for F = 0.5is depicted for systems A and B. The border hDWP betweenthe final configurations (+ − −) and (− − −) is plotted bytriangles. The field hDWP for the Bloch DW is larger than thatfor the Néel DW, which is the same tendency as hNCII and hNC,although the difference is smaller.

The fields hNC and hDWP take close values for E � 0.2,which is similar to those for E � 0.15 in the continuummodel [20] and the anisotropic Heisenberg model [22]. In thecontinuum model and the anisotropic Heisenberg model, hDWP

for F = 0.5 decreases with E . However, we find that hDWP

increases for E > 0.3, and for larger E a higher external fieldis necessary to reverse the magnetization in region I. For largerE , the DW width in the soft magnet becomes shorter. Due tothis narrower DW effect [22,43], the reversed magnetizationin region III is more shielded, which causes less influenceon the unreversed magnetization and the surface nucleationaround the interface. This situation is similar for F = 0.3 atT = 0 in the anisotropic Heisenberg and does not appear inthe continuum model [22]. Considering that F = 0.5(>0.3)is adopted in the present paper, the Nd magnet has narrowerDWs than those in the anisotropic Heisenberg model.

IV. FINITE TEMPERATURE PROPERTIES

We investigate finite temperature effects on the thresholdfields at T = 300 K. Unlike the zero temperature case, nucle-ation and depinning occur stochastically. To obtain hNC, hNCII,and hDWP in the same manner as in the previous work [22],we study a histogram of the number of events for the finalconfigurations.

For the nucleation starting from (+ + +) at field h, thetotal probability for the final configurations is p(+ + +, h) +p(+ − +, h) + p(− − −, h) = 1. If (+ + −) is set as theinitial configuration, the system quickly moves from (+ + −)to (+ − −), and p(+ + −, h) is negligible. Thus, p(+ −−, h) + p(− − −, h) = 1. Here 12 simulations with differentrandom number sequences are performed to obtain the proba-bilities at h.

We estimate the threshold fields hNC, hNCII, and hDWP

as follows. When h increases in the configuration (+ + +),p(+ − +, h) is zero below some h (call h1), and then p(+ −+, h) increases to 1 at another h (call h2). We obtain h1 andh2, and define hNCII as the middle point of this region. Thistransient region is shown by the error bars indicating h1 and h2

for the circles in Fig. 5(a). For larger h, p(− − −, h) increasesfrom 0 to 1 for h3 < h < h4. We define hNC as the middle pointof this region. This point is given by the triangles in Fig. 5(a),in which the error bars indicate h3 and h4. In the same way,hDWP and its error bars are defined in Fig. 5(b).

A. Nucleation

In Fig. 5(a), we depict the threshold fields hNCII between(+ + +) and (+ − +) (circles) and hNC between (+ − +) and(− − −) (triangles) with the comparison between the Blochand Néel DWs at 300 K, and in Fig. 5(b), the threshold fieldshDWP between (+ − −) and (− − −) (triangles) for Bloch andNéel DWs are also given. The threshold fields as functionsof L2 converge or change very slowly at around L2 = 12 aswe see later in Sec. IV C, and thus here we set L2 = 12.We find a stark difference in the region of the thresholdfields between T = 0 K and 300 K (�0.35TC) in the Ndmagnet model. We compare the threshold fields with thoseof the simple anisotropic Heisenberg model [22] for F = 0.5and T = 0.5(�0.35Tc). In the simple anisotropic Heisenbergmodel for 0 � E � 0.5, hNCII and hNC at T = 0 K are at mostfour times larger than those at T = 0.35TC. In the Nd magnetmodel for 0 � E � 0.5, however, we find that at T = 0 KhNCII and hNC take values between 7 T and 26 T, while atT = 300 K they take between 0.8 T and 2.8 T. Namely, hNCII

and hNC at T = 0 K are almost 10 times larger than those

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0

1

2

3

4

0.1 0.2 0.3 0.4 0.5 0.6 0.7

h [T

]

E

Bloch,+++/+-+Bloch,+-+/---

Neel,+++/+-+Neel,+-+/---

0

1

2

3

4

0.1 0.2 0.3 0.4 0.5 0.6 0.7

h [ T

]

E

Bloch, +--/---Neel, +--/---

(a) (b)

FIG. 5. (a) Threshold fields hNCII between (+ + +) and (+ − +) (circles) and hNC between (+ − +) and (− − −) (triangles) with thecomparison between Bloch and Néel domain walls at 300 K. (b) Threshold field hDWP between (+ − −) and (− − −) with the comparisonbetween Bloch (squares) and Néel (crosses) domain walls at 300 K. L2 = 12.

at T = 300 K. In the simple anisotropic Heisenberg model,hDWP at T = 0 K and that at 0.35TC have the same order, butin the Nd magnet model hDWP takes between 11 and 20 T atT = 0 K, while at T = 300 K it takes between 2.2 and 2.5 T.The cause of this large difference between T = 0 K and 300 Kin the Nd magnet model may be related to the competitionbetween the anisotropy terms in the Nd atoms, which is theorigin of the spin-reorientation transition at around 150 K.

The threshold field hNCII for the Bloch DW (blue circles)is larger than that for the Néel wall (green circles), which isdue to the stronger effective exchange interaction along the aaxis, similar to the T = 0 case. The threshold fields hNCII forthe Bloch and Néel DWs increase with E . On the other hand,hNC shows almost constant for E � 0.4. This indicates that asurface nucleation in region I or III is important for furthermagnetization reversal, in which the strength of the magneticanisotropy in regions II is not essential, and that a necessaryfield for the nucleation is almost the same for creations of theBloch and Néel DWs. For E > 0.5, in which the anisotropyenergy in region II is larger than that in regions I and III, hNCII

and hNC coincides and they naturally increase.

B. Depinning

Starting from the initial state (+ + −), the DW in the hardmagnet region III moves quickly to the soft magnet regionII. For relatively small magnetic fields, the magnetization inregion I does not reverse as shown in Fig. 6(a). For highermagnetic fields, the DW in region II moves to the regionI, and finally magnetization reversal completes as shown inFig. 6(b). In Fig. 5(b), the comparison of hDWP between theNéel and Bloch DWs at T = 300 K for L2 = 12 is shown. Thefields hDWP in both DWs are very close and show little E de-pendence (almost constant value) unlike hDWP at T = 0. Thisindicates that the orientation-dependent effective exchangeinteraction is not essential for the depinning. It is consideredthat the effective exchange interaction along the c axes issmaller than that along the a axis in both the soft and hardmagnets and eventually it does not cause the difference ofthe pinning field. It also implies that the anisotropy energyin the soft magnet phase is not so important, and a surfacenucleation in region I around the interface is important for thedepinning.

C. Soft magnet thickness dependence

We study the dependence of the threshold fields on thethickness of the soft magnet. We depict the L2 dependenceof the nucleation field hNC and pinning field hDWP for theBloch and Néel DWs for E = 0.1, 0.3, 0.5, and 0.7 at

0

0.1

0.2

0.3

0.4

0 4 8 12 16 20 24 28 32 36

time

[ns]

x-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5

0

0.1

0.2

0.3

0.4

(a)

(b)

0 4 8 12 16 20 24 28 32 36

time

[ns]

x-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5

FIG. 6. Time evolution of mz along the a axis for E = 0.2 when(a) h = 2.2 T and (b) h = 2.7 T at T = 300 K. x is the position ofthe unit cell along the a axis, located in the middle of the bc plane.The color bars denote the magnitude of the Bohr magneton (μB).

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0

1

2

3

4

5

0 2 4 6 8 10 12 14

h [ T

]

L2

Bloch, nucleationBloch, pinning

Neel, nucleationNeel, pinning

0

1

2

3

4

5

0 2 4 6 8 10 12 14

h [ T

]

L2

Bloch, nucleationBloch, pinning

Neel, nucleationNeel, pinning

0

1

2

3

4

5

0 2 4 6 8 10 12 14

h [ T

]

L2

Bloch, nucleationBloch, pinning

Neel, nucleationNeel, pinning

0

1

2

3

4

5

0 2 4 6 8 10 12 14

h [ T

]

L2

Bloch, nucleationBloch, pinning

Neel, nucleationNeel, pinning

FIG. 7. L2 dependence of nucleation and pinning fields (hNC andhDWP, respectively) for E = 0.1, E = 0.3, E = 0.5, and E = 0.7.Blue circles denote hNC for the Bloch DW and green circles for theNéel DW. Red triangles denote hDWP for the Bloch DW and blacktriangles for the Néel DW.

T = 300 K in Figs. 7(a)–7(d), respectively. The nucleationfields are decreasing functions of L2, while the pinning fieldsare increasing functions of L2.

If the DW width is smaller than L2, it is hard to make a DWstable, which causes the decreasing functions of L2. Indeed,we find that for L2 = 2 and L2 = 3 the (+ − +) configurationdoes not appear as the final state and the transition is alwaysfrom (+ + +) to (− − −). On the other hand, the stabilizationof the DW in a wider region of the soft magnet phase causesthe increasing functions of hDWP against L2.

We find that the value of L2 which gives the convergenceof the nucleation field becomes longer for larger E , while thepining field converges at L2 ∼ 3 and is independent of E . Thissuggests that the anisotropy energy in the soft magnet phase issignificant for the nucleation field, while it is not so importantfor the depinning.

We also find that the difference in hNC between the Blochand Néel DWs is much larger than that in hDWP for L2 beforethe convergence of the fields. The nucleation field hNC forthe Bloch DW is larger than that for the Néel DW, but thepinning field hDWP for the Bloch DW is a bit smaller thanthat for the Néel DW. The orientation-dependent effectiveexchange interaction plays an important role in the gener-ation of the nucleation, but it is not so essential for thedepinning.

V. SUMMARY

We investigated the threshold fields hNCII between (+ + +)and (+ − +), hNC between (+ − +) and (− − −), and hDWP

between (+ − −) and (− − −) in the hard-soft-hard magnetsystem of the Nd magnet based on the atomistic model atT = 0 and 300 K (�0.35Tc) by using the SLLG equation. Weshowed the dependence of the threshold fields on the param-eter E , proportional to the magnetic anisotropy energy of thesoft magnet phase, and the dependence on the width L2 of thesoft magnet phase under the condition of the half strength ofthe exchange interactions in the soft magnet phase, i.e., F =0.5. We also presented the difference in the threshold fieldsbetween the Bloch and Néel DWs, which move along the aand c axes, respectively. We compared the results with thoseof the continuum model at 0 K and of the simple anisotropicHeisenberg model [22] at T = 0 and T = 0.5(�0.35Tc). Wefound that the reduction of the threshold fields due to thethermal fluctuation effect is much larger in the Nd magnetmodel than in the anisotropic Heisenberg model, which is dueto the effects of the competing anisotropy terms in Nd atoms.

At T = 0 K, hNCII, hNC, and hDWP for the Bloch DW arelarger than those for the Néel DW. The origin is the strongereffective exchange interaction along the a (or b) axis than thatalong the c axis [30,33]. E dependence of hDWP at T = 0 Kis different from those of the continuum model [20] and theanisotropic Heisenberg model [22]. The Nd model shows anarrower DW nature, which leads to an increase of hDWP forlarger E .

At T = 300 K for large L2, hNCII for the Bloch DW is largerthan that for the Néel wall, which is also due to the strongereffective exchange interaction along the a axis, similar tothe T = 0 case, but hNC shows no difference and almostconstant except at large E . The configuration (+ + −) doesnot exist stably and changes immediately to (+ − −). Thevalues of hDWP in both DWs are very close and show littleE dependence (almost constant value) unlike hDWP at T = 0.

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The orientation-dependent effective exchange interaction isnot essential for hNC except for large E values and the pinningfield, in which the surface nucleation in the hard magnetregion at around the interface is important.

We also studied the dependence of the threshold fields onthe thickness of the soft magnet L2 with E dependence, andthe following facts were found. The nucleation fields decreasewith L2, while the depinning fields increase with L2. UnlikehNC, L2 at the converged hDWP is almost independent of E .The pinning field converges fast and is constant for L2 longerthan the DW width in the soft magnet. The nucleation fieldhNC for the Bloch DW is much larger than that for the NéelDW before the convergence of the hNC. The anisotropy energyin the soft magnet phase is significant for the nucleationfield, while it is not so important for the depinning, and theorientation-dependent effective exchange interaction plays an

important role in the nucleation but is not essential for thedepinning.

ACKNOWLEDGMENTS

The authors would like to thank Dr. Satoshi Hirosawa foruseful discussion from experimental viewpoints. The presentwork was supported by the Elements Strategy Initiative Centerfor Magnetic Materials (ESICMM) (Grant No. 12016013)funded by the Ministry of Education, Culture, Sports, Sci-ence and Technology (MEXT) of Japan, and was partiallysupported by Grants-in-Aid for Scientific Research C (No.17K05508 and No. 18K03444) from MEXT. Some of thenumerical calculations were performed on the NumericalMaterials Simulator at the National Institute for MaterialsScience.

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