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[Masamichi Nishino](https://orcid.org/0000-0002-2060-2303), Seiji Miyashita

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[Quantitative estimation of coercive field in a ferromagnetic grain using field sweep simulation](https://mdr.nims.go.jp/datasets/f83998ab-4307-4518-b716-3f63e6a6b642)

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Quantitative estimation of coercive field in a ferromagnetic grain using field sweep simulationPHYSICAL REVIEW B 107, 184422 (2023)Quantitative estimation of coercive field in a ferromagnetic grain using field sweep simulationMasamichi Nishino 1,2,3,* and Seiji Miyashita4,5,31Research Center for Advanced Measurement and Characterization, National Institute for Materials Science, 1-2-1 Sengen,Tsukuba, Ibaraki 305-0047, Japan2Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,Tsukuba, Ibaraki 305-0044, Japan3Elements Strategy Initiative Center for Magnetic Materials, National Institute for Materials Science, 1-2-1 Sengen,Tsukuba, Ibaraki 305-0047, Japan4Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan5Physical Society of Japan, 2-31-22 Yushima, Tokyo 113-0033, Japan(Received 18 August 2022; revised 28 March 2023; accepted 1 May 2023; published 10 May 2023)High coercivity is an important property of permanent magnets for application in energy conversion devices.The Nd magnet, Nd2Fe14B, is a typical material. Because coercivity is a long-time relaxation phenomenon,which originates from a strong metastable magnetic state, it is difficult to estimate coercive field (coercive force)studying the time evolution dynamics simulation of a model with atomistic parameters under the limitation ofthe simulation time. In our recent study [M. Nishino et al., Phys. Rev. B 102, 020413(R) (2020)], we presented amethod to estimate coercivity using a statistical method to extend the limitation of simulation time and evaluatedappropriately the coercive field of a single grain of the Nd magnet. In the present study, we propose an alternativemethod to estimate coercivity more conveniently using the field-dependent survival (nonreversal) probabilitygenerated by a time evolution simulation under a field sweep. We demonstrate that the coercive field of the singlegrain can be estimated. In this method, not only coercive field but also the zero-field energy barrier and field forthe zero-energy barrier can be estimated. We discuss detailed features of the estimation of these quantities.DOI: 10.1103/PhysRevB.107.184422I. INTRODUCTIONRealization of high efficiency in energy conversion devicesis a crucial issue for safe energy technology toward sustain-able development goals. High-coercivity permanent magnetssuch as the neodymium (Nd) magnet, Nd2Fe14B [1–9], playan important role. The Nd magnet is used in various electronicdevices, e.g., motors, generators, and compressors, and effortsto increase the coercivity have been performed [10–13]. Coer-cive field Hc is caused by a hysteresis nature of magnets, i.e., anonequilibrium dynamical phenomenon. It depends not onlyon the property of the hard magnet phase but also on thoseof grain boundary, grain shape, etc. [14–22]. Therefore, thecoercivity mechanism is still a difficult issue to be solved.At finite temperatures, the Stoner-Wohlfarth mechanism,i.e., coherent rotation, does not hold in the magnetization re-versal process, but the nucleation process is important becauseit is the trigger of magnetization reversal [23–25]. Nucleationoccurs in a small region, i.e., nm scale, and to understand themicroscopic process of nucleation, recently developed atom-istic models [19,26–41] are quite useful.Unlike continuum modelings developed in micromag-netism [42], in atomistic modelings, the lattice structure(Fig. 1) is introduced with atomic-scale magnetic parameters,estimated from first-principles computation or from experi-*Corresponding author: nishino.masamichi@nims.go.jpmental analyses. Therefore, magnetic properties reflect thedetails of the microscopic structure. Furthermore, atomisticmodelings have another merit. The temperature effect can beanalyzed properly treating all atom spins with the stochasticLandau-Lifshitz-Gilbert (SLLG) equation [43,44] or MonteCarlo (MC) methods, which generate canonical distributionin the equilibrium at a given temperature. Using the atom-istic models, finite-temperature properties of the Nd magnethave recently been investigated. Quantitative analyses on thetemperature dependence of magnetization accompanying aspin-reorientation transition [26–28,36,39], domain wall pro-files [27,30,37,38], surface effects to magnetization reversal[30,40], nucleation features [19,32,33], ferromagnetic reso-nance [31], dysprosium substitution effect [41], etc., havebeen intensively performed.The estimation of coercive field from microscopic informa-tion of magnets is an important subject and various attemptshave been made. It should be noted that the bulk magnet con-sists of many grains and grain boundaries, and the estimationof the coercive field of the bulk is difficult (practically impos-sible) using atomistic models at the present state because oflarge degrees of freedom. To study fundamental informationof the coercivity, the coercive field of a single grain has beenestimated based on an atomistic model [32,33].Coercive field has been studied from the viewpoint of afree-energy barrier. The minimum energy path method hasbeen used to obtain the change of the free energy along a pathof evolution of magnetization from a metastable to stable state2469-9950/2023/107(18)/184422(10) 184422-1 ©2023 American Physical Societyhttps://orcid.org/0000-0002-2060-2303http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevB.107.184422&domain=pdf&date_stamp=2023-05-10https://doi.org/10.1103/PhysRevB.102.020413https://doi.org/10.1103/PhysRevB.107.184422MASAMICHI NISHINO AND SEIJI MIYASHITA PHYSICAL REVIEW B 107, 184422 (2023)FIG. 1. Unit cell of Nd2Fe14B. Blue, red, and yellow balls denote Fe, Nd, and B atoms, respectively.[45,46]. For atomistic model studies, the angular dependenceof the free energy of the hard magnet phase of the Nd magnetwas estimated [26] using the constrained Monte Carlo method[47]. The free energy of a single grain of the Nd magnet as afunction of the field was obtained from the concept of thermalactivation [33].Coercivity is a phenomenon of a nonequilibrium long-timerelaxation process, and to approach the mechanism of coerciv-ity, time evolution dynamics analyses are important. However,there exists a difficulty in time evolution dynamics simula-tions, i.e., simulation time. Experimentally coercive field isdefined as a field at which the relaxation time is 1 s. On theother hand, a practical simulation time is around 1 ns, andit is too short to study such a long-time relaxation process.Simulation time is a common problem to studies on long-time relaxation phenomena in all real systems, e.g., biologicalsystems.Recently, we proposed a method to estimate the coercivefield to extend the limitation of simulation time in Ref. [32]. Inthis method, first, we performed a simulation of magnetizationreversal for many samples in a fixed time period (t < 1 ns)under a fixed field, and observed the survival (unrelaxed)probability as a function of time (t ). Then, using a statisticalrelation to the probability, we determined the relaxation time.There the relaxation time could be estimated up to microsec-onds or submicroseconds. Finally, we estimated the field forthe 1 s relaxation time by an extrapolation of the relaxationtime. Using this method, we estimated the coercive field of asingle grain to be Hc � 3.0–3.2 T. This was consistent withthe value estimated for the same grain by a MC study [33]with the Wang-Landau algorithm [48]. In this MC study, thefield dependence of the free-energy barrier was comparedto �F corresponding to 1 s Arrhenius relaxation time, τ =τ0 exp(β�F ), where τ0 is a prefactor and β is the inversetemperature.The above-mentioned two methods require very heavycomputational costs with complexity. For the study of co-ercivity in various situations, we need a more convenientmethod to obtain the coercivity. In the present study we showan alternative convenient method to estimate coercivity. Wedemonstrate that the coercive field of the same single graincan be estimated with high accuracy using a field sweep (lessthan 1 ns).There are three kinds of the (free) energy shape for magne-tization reversal as shown in Fig. 2, i.e., barrier crossing type,marginal type, and no-barrier type, which lead to stochas-tic, intermediate, and deterministic dynamics, respectively. Incontrast to the experiments, only a short-time measurement isallowed in a time evolution dynamics simulation. Therefore,in the present paper, we consider the survival (nonreversal)probability extending the field sweep range from the stochas-tic region to the deterministic region whose relaxation timeis much shorter than the stochastic one. We show that thecoercive field is estimated with high accuracy and also presentthe estimation of the zero-field energy barrier and field for thezero-energy barrier, which are important for the analysis ofthe metastable property.The rest of the paper is organized as follows. In Sec. II, theatomistic model for the Nd magnet is explained. In Sec. III,the time evolution dynamics method to estimate the coercivityis presented. In Sec. IV, the results and discussion are given.Section V is devoted to the summary.II. MODELWe adopt the following atomistic Hamiltonian for the Ndmagnet:H = −∑i< j2Ji jsi · s j −Fe∑iDi(szi)2+Nd∑i∑l,m�l,iAml,i〈rl〉iÔml,i − H∑iSzi . (1)FIG. 2. Typical energy barrier types for magnetization reversal.(a) Barrier crossing type, (b) marginal type, and (c) no-barrier type,which lead to stochastic, intermediate, and deterministic dynamics,respectively.184422-2QUANTITATIVE ESTIMATION OF COERCIVE FIELD IN … PHYSICAL REVIEW B 107, 184422 (2023)Here, Ji j is the exchange interaction between the ith and jthsites, and Di is the magnetic anisotropy constant for Fe atoms.The third term is the crystal electric field energy of Nd atoms,and �l,i, Aml,i, 〈rl〉i, and Ôml,i are the Stevens factor, coefficientof the spherical harmonics of the crystalline electric field, av-erage of rl over the radial wave function, and Stevens operator,respectively. We consider l = 2, 4, 6 and m = 0 (diagonal op-erators), which provide the dominant contribution. The fourthterm is the Zeeman term, and H is the external magnetic field.For Fe and B atoms, si denote the magnetic moment at the ithsite, while for Nd atoms, it is the moment of the valence (5dand 6s) electrons. The total moment for Nd atoms at the ithsite is Si = si + J i, where J i = gTJiμB with the magnitudeof the total angular momentum, J = 9/2, and Landé g factor,gT = 8/11. We define si = si for Fe and B atoms.The details of the model are given in our previous pa-pers [26–28,39], in which the magnetic interactions weremainly obtained from first-principles computation methods.We showed the spin-reorientation transition temperature, Tr =150 K, which is close to the experimentally estimated tem-perature [7–9,49], and the critical temperature, Tc ∼ 870 K,which is a little overestimated from the experimental valuesTc ∼ 600 K [4,7], due to an overestimation of the exchangeinteractions. We are interested in room temperature propertiesand set T = 400 K � 0.46Tc, which is close to room temper-ature practically.III. DYNAMICAL METHODA. Real time dynamics with thermal fluctuation effectWe employ the SLLG equation [43,44] to study the timeevolution dynamics of the Nd magnet:ddtSi = − γ1 + α2iSi × (Heffi + ξi)− αiγ(1 + α2i)SiSi × [Si × (Heffi + ξi)]. (2)Here αi is the Gilbert damping factor at the ith site and γis the gyromagnetic constant. Heffi = − ∂H∂Siis the effectivefield applied at the ith site from the exchange interactions,anisotropy terms, and Zeeman term, and ξi(t ) = (ξ xi , ξyi , ξ zi )is a white-Gaussian noise field with the following properties:〈ξμi (t )〉 = 0,〈ξμi (t )ξνj (s)〉 = 2Diδi jδμνδ(t − s). (3)The temperature of the system, T , is a function of thestrength of the random noise field Di according to the fluc-tuation dissipation relation:Di = αiSikBTγ. (4)When this relation is satisfied, the system relaxes to the equi-librium state in the canonical distribution at temperature T .The value of αi is unknown for the Nd magnet, and we assumeαi = α = 0.1, which is a typical value for magnets [42].We apply a kind of middle-point method [44] equivalentto the Heun method [43] for the numerical integration of thestochastic differential equation in the Stratonovich interpreta-tion. For the time step of this equation, �t = 0.1 fs is used.B. Survival probability under field sweepFor a weak reversed magnetic field, a magnetization rever-sal occurs in a barrier crossing process as shown in Fig. 2(a),which leads to a stochastic dynamics. The relaxation rate forthe stochastic dynamics is given by the Arrhenius rate asR = 1τ0e−βEB(H ), (5)where τ0 is a preexponential factor, which represents of thefrequency of the contact with the bath and is of order of thelattice vibration frequency. We adopt a commonly used valuefor the factor, i.e., τ0 = 10−11 s [23], which was used in thepaper giving the reference data [33].On the other hand, for a strong reversed field, a magnetiza-tion reversal occurs deterministically as shown in Fig. 2(c).For the deterministic process, the relaxation rate is set toconstant:R = const. (6)The relaxation rate should vary with the field even in the deter-ministic region, but it is much faster than that in the stochasticregion, and thus the choice is not relevant to estimate thecoercive field.For an intermediate field, the dynamics has a crossoverfeature [Fig. 2(b)]. In the present study, we consider an ap-proximate form of relaxation rate which describes both thestochastic and deterministic regions including the crossover(intermediate) field region:R = 1τ0(eβEB(H ) + c), (7)which satisfies R � 1τ0e−βEB(H ) for positive large values ofEB(H ) corresponding to the stochastic region, and R �1/(τ0c) for small values of EB(H ) corresponding to the de-terministic region. Using this probability R, we derive theprobability of nonrelaxation when the field is swept until Hstarting from H = −∞ (all down state) as follows.Since the relaxation rate at time t is R(H (t )), if a magne-tization reversal does not occur until time t , the probabilityof avoiding magnetization reversal at t + �t is given as 1 −R(H (t ))�t = e−R(H (t ))�t + O(�t2). Then, the probability ofavoiding magnetization reversal during the time [t0, t], where�t = (t − t0)/n, is described asP(t ) = [1 − R(H (t0))�t][1 − R(H (t0 + �t ))�t]× [1 − R(H (t0 + 2�t ))�t] · · · [1 − R(H (t − �t ))�t]= e−R(H (t0 ))�t e−R(H (t0+�t ))�t · · · e−R(H (t−�t ))�t + O(�t )=n−1∏m=0[e−R(H (t0+m�t ))�t ] + O(�t ). (8)For n → ∞ (�t → 0) and t0 → −∞, the probability is givenasP(t ) = exp[− 1τ0∫ t−∞1eβEB(H (t ′ )) + cdt ′]. (9)184422-3MASAMICHI NISHINO AND SEIJI MIYASHITA PHYSICAL REVIEW B 107, 184422 (2023)If the field is swept linearly with time, i.e., H (t ) = vt , thenthe probability as a function of H is given asP(H ) = exp[− 1τ0∫ H (t )H (−∞)=−∞1eβEB(h) + cdtdhdh]= exp[− 1vτ0∫ H−∞1eβEB(h) + cdh]. (10)In the present work, we use the following formula for the(free) energy barrier:EB(H ) = E0(1 − HH0)n, (11)where E0 is the zero-field energy barrier and H0 is the fieldfor zero-energy barrier. This formula has often been employedfor studies of magnetization reversal in permanent magnets[21,50,51].The value of the exponent n is established as n = 1 ∼ 2for many magnetic materials. For coherent magnetic reversalas in the Stoner-Wohlfarth model, the exponent is n = 2 [52].Givord et al. studied a thermally activated magnetization re-versal using n = 1 based on an experimental observation offield independence in the fluctuation field, Sv = kBT/( ∂EB∂H )T ,for several Nd-Fe-B magnets [23,24]. n = 1 was also sug-gested for weak domain pinning [53], and then n = 1 wasexperimentally observed as the domain wall pinning processin several magnets [54,55].Recently, n � 1 for several Nd-Fe-B magnets was exper-imentally determined for a nucleation process [21,51] usinga magnetic viscosity measurement and the Sharrock equa-tion [56,57] (relation of coercivity vs reversal time). n � 1was also confirmed by an experiment observing the reversalprobability against a field sweep [51]. n = 1 was theoreticallysuggested in a recent nucleation model study [58] and in aMonte Carlo study for the Nd magnet atomistic model [33].Therefore, in the present study, we adopt n = 1 primarily. Wealso investigate the case of n = 2 in the Appendix and discussthe difference between the two cases.For EB(H ) = E0(1 − HH0), i.e., the n = 1 case, P(H ) isgiven asP(H ) = exp[− 1vτ0∫ H−∞1eβE0(1− hH0) + cdh](12)= exp[− 1vτ0e−βE0∫ H−∞1e− βE0H0h + ce−βE0dh].(13)Using the relation∫dxλ + eκx= 1λκ(κx − ln |λ + eκx|) (14)withλ = ce−βE0 , κ = −βE0H0, (15)we haveP(H ) = exp[− H0cvτ0βE0ln(1 + ce−βE0 (1− HH0))]. (16)Then we haveln{− ln[P(H )]} = ln[H0cvτ0βE0ln(1 + ce−βE0 (1− HH0))].(17)For c → 0, i.e., the pure Arrhenius case,ln{− ln[P(H )]} = βE0H0H + ln[e−βE0H0vτ0βE0], (18)which is a linear function of the field H . Therefore, a lineardependence of H for ln{− ln[P(H )]} indicates that a stochas-tic dynamics is realized. On the other hand, when the functionof ln{− ln[P(H )]} deviates from a linear dependence of H ,it suggests that the dynamics is not the stochastic one butintermediate or deterministic one.Applying the formula (17) or (18) to time evolution simula-tion under a field sweep, we estimate H0 and βE0 (optimizedvalues). From the values of H0 and βE0, we obtain the co-ercive field Hc as follows. The relaxation time is given asτ = τ0 exp(β�F ) for the free-energy barrier �F , and thecoercive field is defined as the threshold field at which therelaxation time is 1 s. Thus, β�F = 25.3 for τ = 1 s andτ0 = 10−11 s. From the relationβF = βE0(1 − HcH0)= 25.3, (19)the coercive field is given as a function of H0 and βE0, i.e.,Hc = H0(1 − 25.3βE0). (20)We investigate an open-boundary system of 12 × 12 × 9unit cells (10.56 nm × 10.56 nm × 10.971 nm) along the a,b, and c axes, respectively. It has been confirmed that nucle-ation occurs from a corner at T = 0.46Tc in similar systemsizes [33,59] including this size [32], and the dipole-dipoleinteraction has negligible effect in this size [39]. First, Nsamples with a down-spin state are prepared, and magneti-zation reversal under a magnetic field sweep is observed. Themagnetic field is swept with a constant velocity (v) from a lowfield (Hi) to a high field (Hf ). For each sample, the per-sitemagnetization,m = 1NsiteNsite∑i=1Szi , (21)where Nsite is the number of the atoms in the system, iscomputed as a function of H . We determined the reversal fieldas the field when the value of m changes from negative topositive. During the field sweeping, the number of the survival(nonrelaxed) samples, Ns(H ), is counted as a function of H .P(H ) is estimated as P(H ) = Ns (H )N . Then, ln{− ln[P(H )]}is plotted vs H . A demonstration will be given in the nextsection. Equation (18) or (17) is used as a fitting function forthis plot, where βE0 and H0 are fitting parameters for Eq. (18)and βE0, H0, and c are those for Eq. (17).IV. ESTIMATION OF COERCIVE FIELDWe study magnetization reversal under a field sweep fromH = 3.7 T to H = 4.2 T for 0.5 ns (v = 1 × 109 T/s). We184422-4QUANTITATIVE ESTIMATION OF COERCIVE FIELD IN … PHYSICAL REVIEW B 107, 184422 (2023)(a)(b)FIG. 3. (a) Magnetization, m, and (b) survival probability, P, atv = 1 × 109 T/s under a field sweep from H = 3.7 T to 4.2 T. N =1536. Ns = 1436 at H = 4.2 T.simulate magnetization reversal using N = 1536 samples withdifferent random number sequences for the noise field. Mag-netization reversal curves (m) and survival probability (P(H ))are plotted as a function of the field H in Fig. 3(a) andFig. 3(b), respectively. We find that the interval of the reversalfield is distributed sparsely and at higher fields the frequencyof relaxation increases. In Fig. 4, we find that for 4.07 T� H � 4.2 T, ln{− ln[P(H )]} shows a rather well definedregion with a linear dependence on H . These observationssuggest that the relaxation occurs stochastically for 4.07 T� H � 4.2 T. For H � 4.07 T, however, we find a deviationfrom the linear dependence. This region suggests the exis-tence of an initial transient process before a regular relaxationprocess, which we also encountered in the observation ofmagnetization relaxation with time dependence for a fixedfield in the previous study. This range should be excluded fromthe coercivity analysis.Using the function (18), we perform a least-squares fitto the data in the region of 4.07 T � H � 4.2 T, which isshown by the line in Fig. 4. The intercept on the verticalaxis ln{− ln[P(H )]}, which corresponds to ln[e−βE0 H0vτ0βE0],and the slope of the fitting line, which corresponds to βE0H0,permit calculation of the fitting parameters as H0 = 4.46 TFIG. 4. ln{− ln[P(H )]} (red circles) under a field sweep at v =1 × 109 T/s from H = 3.7 T to 4.2 T. Solid line represents the resultof fitting with Eq. (18) (see text for details).and βE0 = 76.0. Thus, using the formula (20) with these H0and βE0, the coercive field is obtained as Hc = 2.98 T. Wefind that this value is very close to the previous estimation ofthe coercive field (Hc � 3.0–3.2 T) in the same grain, whichindicates the capability of the present method to reproduce acompatible result.If we adopt a different fitting range, e.g., 4.02 T � H �4.2 T, the estimated coercive field Hc = 3.05 T is even closerto our anterior result of Hc � 3.0–3.2 T: the fitting values forthis case are H0 = 4.44 T and βE0 = 81.1. Even if the fittingrange is chosen as 0.3958 T � H � 4.2 T, which includes thefirst relaxation point and the initial transient process (inade-quate choice), we find that the estimated coercive field, Hc =3.10 T, is also close to Hc = 2.98 T. The estimated values forHc, H0, and βE0 for different fitting ranges are summarized inTable I. We find that the values of Hc and H0 do not changemuch depending on the fitting range. Because a rather welldefined linear region exists, these fittings in different rangesgive similar values. The difference in the values gives uncer-tainty of the present estimation, which is not very small, butthe estimated values are in an acceptable range.Next, we investigate magnetization relaxation under afaster field sweep (v = 1.4 × 109 T/s) in a wider field region(from H = 3.8 T to H = 4.5 T for 0.5 ns) for N = 1536. InFig. 5(a) and Fig. 5(b), magnetization reversal curves (m) andsurvival probability [P(H )] are presented, respectively, as afunction of the field H . Here, the number of nonrelaxed sam-TABLE I. Estimated values of Hc, H0, and βE0 using Eq. (18) inthe fitting range between Hi and Hf for a field sweep at v = 1 × 109T/s from H = 3.7 T to 4.2 T. The field values are given in teslas. Thevalues in parentheses are standard errors.[Hi, Hf ] Hc H0 βE0[4.07, 4.2] 2.98(2) 4.461(4) 7.60(7)×101[4.02, 4.2] 3.05(2) 4.437(4) 8.11(9)×101[3.958, 4.2] 3.10(2) 4.423(5) 8.44(10)×101184422-5MASAMICHI NISHINO AND SEIJI MIYASHITA PHYSICAL REVIEW B 107, 184422 (2023)(b)(a)FIG. 5. (a) Magnetization, m, and (b) survival probability, P,under a field sweep at v = 1.4 × 109 T/s from H = 3.8 T to 4.5 T.N = 1536. Ns = 728 at H = 4.5 T.ples (Ns = 728) is smaller than that illustrated in Fig. 3 (Ns =1436), and relaxation data points are packed more densely atthe higher field values (0.43 T � H).In Fig. 6, ln{− ln[P(H )]} is depicted as a function of H . Wediscern three characteristic regions in this plot, i.e., region I:the initial transient region for 4.1 T � H � 4.15 T, in whichthe data deviate from the dependence shown by the line forEq. (17); region II: a linear dependence for 4.15 T � H � 4.3T; and region III: a bending curve for 4.3 T � H � 4.5 T. Re-gion II suggests a stochastic relaxation region [Fig. 2(a)], andregion III indicates an intermediate [Fig. 2(b)] or deterministicregion [Fig. 2(c)].In this case, we use Eq. (17) with three fitting parametersH0, βE0, and c. We perform a least-squares fitting to the datain the region of 4.15 T � H � 4.5 T. We obtain the coer-cive field Hc = 3.11 T, and optimized values: H0 = 4.49 T,βE0 = 82.6, and c = 17.3. If the fitting rage is shifted as4.12 T � H � 4.5 T, Hc = 3.19 T and H0 = 4.47 T are ob-tained, and the estimated values of the coercive field Hc andoptimized H0 hardly change. The relative percentage of thechange for Hc with the values of 3.11 and 3.19 is (3.19 −3.11)/3.11 × 100% = 2.57%, and that for H0 with the valuesof 4.49 and 4.47 is (4.49 − 4.47)/4.49 × 100% = 0.445%.Even if the fitting range is changed as 4.1 T � H � 4.5 TFIG. 6. ln{− ln[P(H )]} (red circles) under a field sweep at v =1.4 × 109 T/s from H = 3.8 T to 4.5 T. Solid curve represents theresult of fitting with Eq. (17) (see text for details).(unphysical choice), which includes the first relaxation point,the estimated values are Hc = 3.26 T and H0 = 4.45 T, whichare close to the values obtained for 4.15 T � H � 4.5 T.Estimated values are summarized in Table II. We also find thatthe values of Hc and H0 are not much affected by the choiceof the fitting region.As a reference to the estimation using Eq. (17), we alsoperform a least-squares fit using Eq. (18) for this data, andcompare the results. The fitted line is given in Fig. 7 in the fit-ting range of 4.15 T � H � 4.3 T. We find that the estimatedvalues, Hc = 3.09 T, H0 = 4.50 T, and βE0 = 80.7, are veryclose to those estimated applying Eq. (17). Estimated valuesin different fitting ranges are given in Table III. We again findthat the estimated Hc and H0 are almost the same between thetwo methods using Eqs. (18) and (17). Compared to Hc andH0, βE0 is more sensitive to the fitting range.We estimated Hc, H0, and βE0 in two different sweepings.P(H ) reduces to 0.935 from H = 3.7 to 4.2 T, while P(H )reduces to 0.474 from H = 3.8 to 4.5 T. The former is a slowsweeping case, while the latter is a fast sweeping case. Al-though region III corresponding to the deterministic relaxationappears in the fast sweeping, the values of Hc, H0, and βE0 inthe fast sweeping are still estimated to be close to those in theslow sweeping.However, in faster sweeping, the region II shrinks (regionIII dominates) and it becomes difficult to estimate the coercivefield. It gives the limitation of applicability of the presentmethod.TABLE II. Estimated values of Hc, H0, and βE0 using Eq. (17) inthe fitting range between Hi and Hf for a field sweep at v = 1.4 × 109T/s from H = 3.8 T to 4.5 T. The field values are given in teslas. Thevalues in parentheses are standard errors.[Hi, Hf ] Hc H0 βE0 c[4.15, 4.5] 3.11(1) 4.486(2) 8.26(5)×101 1.73(2)×101[4.12, 4.5] 3.19(2) 4.466(3) 8.83(7)×101 1.92(4)×101[4.10, 4.5] 3.26(2) 4.447(3) 9.45(10)×101 2.11(5)×101184422-6QUANTITATIVE ESTIMATION OF COERCIVE FIELD IN … PHYSICAL REVIEW B 107, 184422 (2023)FIG. 7. ln{− ln[P(H )]} (red circles) under a field sweep at v =1.4 × 109 T/s from H = 3.8 T to 4.5 T. Solid line represents theresult of fitting with Eq. (18) (see text for details).From the above analyses, we obtain Hc � 3.0–3.1 T, H0 �4.4–4.5 T (Hc/H0 � 0.67–0.69), and βE0 � 76–83. For T =0.46Tc = 400 K in the simulation, E0/kB � 30 400–33 200 K.We found that the value of Hc is very close to that estimated bythe previously developed method, i.e., Hc � 3.0–3.2 T. In thepresent method, we need not to obtain the relaxation times forseveral values of the field. Thus, we conclude that the presentmethod can estimate coercivity approximately with much lesseffort compared to the previous one. Furthermore, this methodcan evaluate not only Hc but also H0 and E0β, which is anothermerit of this method.In the present work, we used the formula of the potentialbarrier: EB(H ) = E0(1 − HH0)n with n = 1 using Eqs. (17) and(18), which produced fitting results with good precision. Aswas mentioned in the Introduction, the discussion about theproper choice of n is not settled yet. Here we show that n = 1is suitable for simulation of nucleation-triggered magnetiza-tion reversal in a Nd magnet.Following the Introduction section, let us recall that co-ercive field is changed, in addition to physical properties ofindividual grains, also by conditions of grain boundaries andsome other parameters. Quantitative comparison with exper-imentally estimated coercive fields is out of the scope of thepaper, because our objective is to estimate the coercive fieldof a single grain while experimental systems are assemblies ofgrains. Here, we just introduce experimental situation of coer-cive fields of Nd2Fe14B magnets. References [21,51] reportedthat Hc, H0, and E0/kB were estimated for 25–200 ◦C in twotypes of hot-deformed Nd-Fe-B magnets: an as-hot-deformedTABLE III. Estimated values of Hc, H0, and βE0 using Eq. (18)in the fitting range between Hi and Hf for a field sweep at v = 1.4 ×109 T/s from H = 3.8 T to 4.5 T. The field values are given in teslas.The values in parentheses are standard errors.[Hi, Hf ] Hc H0 βE0[4.15, 4.3] 3.09(1) 4.499(2) 8.07(5)×101[4.12, 4.34] 3.07(2) 4.508(4) 7.95(9)×101[4.1, 4.4] 2.95(3) 4.548(4) 7.22(9)×101(HD) magnet and a Nd-Cu eutectic alloy grain-boundary dif-fused (GBD) magnet. Estimated values at 25◦C � 0.5Tc inexperiments [21] are Hc = 1.1 T for HD and Hc = 2.2 T forGBD, Hc/H0 = 0.78 for HD and Hc/H0 = 0.86 for GBD, andE0/kB = 40 000 K for HD and E0/kB = 60 000 K for GBD.Although the values of Hc in experiments are smaller due tothe ensemble effects, we consider that our estimation maybe a good reference for further development of studies oncoercive field. Indeed, the coercive field value depends onsample preparation technique and can reach Hc � 3 T [60].V. SUMMARYWe propose a new method for coercive field estimation,which is robust and convenient, because it does not requireample magnetization simulations reaching long observationtimes characteristic to the experimental studies. Our resultscan be obtained faster, and moreover, the algorithmic imple-mentation of this new method can benefit considerably fromdistributed calculations and parallel computing.Using this method, we estimated the coercive field of asingle grain of the Nd magnet. Depending on the field-sweepregion, the feature of dynamics varies as stochastic, crossover,and deterministic ones at lower, middle, and higher fieldregions, respectively. This method works well if the sweeprange includes the stochastic relaxation region. When the fieldsweep is within the stochastic region, Eq. (18) is available,while when it spans over the stochastic region to deterministicregion, Eq. (17) is available.In general, the reversal frequency is low in the stochasticregion, which tends to lead to insufficient sampling to P(H ),and the use of Eq. (17) may be practical for faster relaxation infaster sweeping. In addition to the estimation of coercive fieldHc, this method provides the estimation of zero-field energybarrier βE0 and field for zero-energy barrier H0, which areimportant indices in experimental analyses of coercivity.When we perform a fitting using Eq. (18) or Eq. (17) toestimate coercivity, the relaxation data in the initial transientprocess before the regular relaxation in the stochastic regionof the field should be excluded. However, exact identificationof the border between the initial transient and regular relax-ation regions is not necessary, because the values of coercivefield and field for zero-energy barrier are not much affected bythe fitting range, and it is possible to estimate approximatelyHc and H0. The zero-energy barrier βE0 is also well estimated,although it is more sensitive to the fitting range.We used EB(H ) = E0(1 − HH0)n assuming n = 1 for the po-tential barrier in the formulation of Eqs. (17) and (18). Indeed,the estimated Hc using n = 1 was closer to the previouslyestimated one than using n = 2, and the adoption of n = 1was found to be valid for the estimation of Hc.ACKNOWLEDGMENTSThe authors would like to thank Prof. Okamoto and Dr.Hirosawa for useful discussions about experimental estima-tions of coercive field, zero-field energy barrier, and fieldfor zero-energy barrier, and experimental features of the Ndmagnet. The present work was supported by the ElementsStrategy Initiative Center for Magnetic Materials (ESICMM)184422-7MASAMICHI NISHINO AND SEIJI MIYASHITA PHYSICAL REVIEW B 107, 184422 (2023)(Grant No. 12016013) funded by the Ministry of Education,Culture, Sports, Science, and Technology (MEXT) of Japan,and was partially supported by Grants-in-Aid for ScientificResearch C (No. 18K03444 and No. 20K03809) from MEXT.The numerical calculations were performed on the NumericalMaterials Simulator at the National Institute for MaterialsScience.APPENDIX: ESTIMATIONOF COERCIVE FIELD USING n = 2In the present study, we adopted n = 1 for Eq. (11). Here,we mention the adoption of n = 2. In this Appendix, we showthe estimation of Hc, H0, and βE0 using n = 2.If EB(H ) = E0(1 − HH0)2 is adopted, the survival probabil-ity isP(H ) = exp[− 1vτ0∫ H−∞1eβE0(1− hH0)2 + cdh]. (A1)It is difficult to obtain analytical solution for the integral (A1),so we consider a simplified case with c = 0.Using the relation∫ H−∞e−βE0(1− hH0)2dh =( ∫ H0−∞dh +∫ HH0dh)e− βE0H20(h−H0 )2=∫ 0−∞e− βE0H20x2dx + H0√βE0∫ X00e−x2dx= 12√πβE0H0[1 + erf (X0)], (A2)where X0 =√βE0H0(H − H0), we haveP(H ) = exp[− 12vτ0√πβE0H0[1 + erf (X0)]]. (A3)Then, we obtainln{− ln[P(H )]} = ln12vτ0√πβE0H0[1 + erf (X0)]. (A4)FIG. 8. ln{− ln[P(H )]} (red circles) under a field sweep at v =1 × 109 T/s from H = 3.7 T to 4.2 T. Solid curve represents theresult of fitting with Eq. (A4) (see text for details).TABLE IV. Estimated values of Hc, H0, and βE0 using Eq. (A4)in the fitting range between Hi and Hf for a field sweep at v = 1 ×109 T/s from H = 3.7 T to 4.2 T. The field values are given in teslas.The values in parentheses are standard errors.[Hi, Hf ] Hc H0 βE0[4.07, 4.2] 3.34(2) 4.830(8) 2.67(5)×102[4.02, 4.2] 3.39(2) 4.790(8) 2.94(6)×102[3.958, 4.2] 3.40(2) 4.775(8) 3.05(6)×102Once H0 and βE0, i.e., fitting parameters, are obtained, Hcis estimated in the same manner as the derivation of Eq. (20).From the relationβF = βE0(1 − HcH0)2= 25.3, (A5)the coercive field is given as a function of H0 and βE0, i.e.,Hc = H0(1 −√25.3βE0). (A6)In Fig. 8, we perform a least-squares fit using Eq. (A4)to the data of ln{− ln[P(H )]} in the region of 4.07 T � H �4.2 T, studied in Fig. 4. We find that the function Eq. (A4)shows a gently bending curve. The estimated Hc, H0, and βE0are 3.34 T, 4.83 T, and 267, respectively. Those in differentfitting ranges are summarized in Table IV. In Fig. 9, wealso perform a least-squares fit using Eq. (A4) to the data ofln{− ln[P(H )]} in the region of 4.15 T � H � 4.3 T, inves-tigated in Fig. 6. The estimated Hc, H0, and βE0 are 3.48 T,4.81 T, and 329, respectively. Those in different fitting rangesare summarized in Table V. The estimated values of Hc andH0 are not much affected by the fitting region, which is thesame tendency as the n = 1 case.We find that the estimated Hc using Eq. (18) or Eq. (17) iscloser to the previously estimated Hc than that using Eq. (A4),although that using Eq. (A4) slightly overestimates Hc.FIG. 9. ln{− ln[P(H )]} (red circles) under a field sweep at v =1.4 × 109 T/s from H = 3.8 T to 4.5 T. Solid curve represents theresult of fitting with Eq. (A4) (see text for details).184422-8QUANTITATIVE ESTIMATION OF COERCIVE FIELD IN … PHYSICAL REVIEW B 107, 184422 (2023)TABLE V. Estimated values of Hc, H0, and βE0 using Eq. (A4)in the fitting range between Hi and Hf for a field sweep at v = 1.4 ×109 T/s from H = 3.8 T to 4.5 T. 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