Grain-size dependent demagnetizing factors in

permanent magnets

S. Bance1, B. Seebacher1, T. Schrefl2, L. Exl3, M. Winklhofer4, G.Hrkac5, G. Zimanyi6, T. Shoji7, M. Yano7, N. Sakuma7, M. Ito7, A.

Kato7, and A. Manabe7

1Department of Technology, St Polten University of Applied Sciences,Matthias Corvinus-Str. 15, 3100 St Polten, Austria

2Center for Integrated Sensor Systems, Danube University Krems,2700 Wiener Neustadt, Austria

3Solid State Physics, Vienna University of Technology, 1040 Vienna,Austria

4Department of Earth and Environmental Science,Ludwig-Maximilians-Universitat Munchen, 80333 Munich, Germany4Fakultat fur Physik, Universitat Duisburg-Essen, 47048 Duisburg,

Germany5CEMPS, Harrison Building, University of Exeter, Exeter, EX4 4QF,

UK6Department of Physics and Astronomy, UC Davis, One Shields

Avenue, Davis, California 95616, USA7Toyota Motor Corp., Toyota City, 471-8572, Japan

3 October 2014

The following article appeared in S. Bance et al., “Grain-Size Dependent Demagne-tizing Factors in Permanent Magnets.“ J. App. Phys. vol. 116 no. 23 pp. 233903 and maybe found at http://scitation.aip.org/content/aip/journal/jap/116/23/10.1063/1.4904854

©2014 American Institute of Physics. This article may be downloaded for personal useonly. Any other use requires prior permission of the author and the American Instituteof Physics.

email: s.g.bance@gmail.comweb: http://academic.bancey.com

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Abstract

The coercive field of permanent magnets decreases with increasinggrain size. The grain size dependence of coercivity is explained by asize dependent demagnetizing factor. In Dy free Nd2Fe14B magnetsthe size dependent demagnetizing factor ranges from 0.2 for a grainsize of 55 nm to 1.22 for a grain size of 8300 nm. The comparison ofexperimental data with micromagnetic simulations suggests that thegrain size dependence of the coercive field in hard magnets is due tothe non-uniform magnetostatic field in polyhedral grains.

1 Introduction

Renewable energy technologies heavily rely on permanent magnets.[1] Per-manent magnets are used in direct drive wind power generators and in themotors and generators of hybrid vehicles. Modern high performance perma-nent magnets are based on Nd2Fe14B. In many applications the magnets areused at temperatures well above room temperature. For example the oper-ating temperature in hybrid vehicle applications is around 450 K. In orderto maintain the required coercive field at this temperature neodymium (Nd)is partially replaced by heavy rare earth elements such as dysprosium (Dy).In the quest for reduction of critical elements including heavy rare earthselements methods for improving the coercive field of Nd2Fe14B magnetswithout Dy addition are sought. One of the strategies for high coercivityDy-free magnets is the reduction of grain size. In order to produce alignedfine-grained, high coercivity magnets press-less sintering[2] or hot-pressingof melt-spun ribbons with (Nd,Cu) infiltration[3] have been introduced.

It is a well-known experimental fact that the coercive field of permanentmagnets increases with decreasing grain size, however, there is little theoret-ical understanding for this. This has to be attributed to the various factorswhich influence the reversal processes that change together with the grainsize. The equation[4]

Hc(T ) = αHA(T )−NeffMs(T )−Hth(T ) (1)

may be used to analyze magnetization reversal in permanent magnets. HereHA is the anisotropy field and Ms is the saturation magnetization. Themicrostructural parameter α accounts for the reduction of the coercive fieldHc owing to defects and misalignment. The effective demagnetizing factorNeff accounts for the reduction of Hc by local magnetostatic interactioneffects. It is, in general, not identical to the geometric demagnetizing factorand can be greater than 1. Hth includes the negative effects on coercivityowing to thermal fluctuations. The parameters α and Neff , which can beused to categorize permanent magnets, are extracted from the temperaturedependent values of Hc(T ), HA(T ), and Ms(T ). This is done by plottingHc(T )/Ms(T ) versus HA(T )/Ms(T ) and fitting a straight line. When the

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thermal fluctuation field is not explicitly taken into account in the analysis,Neff , which is obtained from the intercept with the Hc/Ms-axis, also containsa term Hth/Ms.

In this work, the grain size dependence of the coercive field is inves-tigated. In particular the focus is on Dy-free magnets. In hot-deformedmagnets with (Nd,Cu) infiltration[3] it is possible to keep α constant and tocontrol Neff . When selecting magnets for our experimental study of the grainsize dependence of coercivity, we choose magnets with a similar α value.Therefore major differences in the coercive field between the investigatedmagnets have to be attributed to changes in the effective demagnetizationfactor due to local magnetostatic field effects. By comparing experimentaland computational results we show how the empirical relationship betweencoercive field and grain size can be understood using the theory of micro-magnetism. In order to compare the results with experimental data wecompute a grain size dependent demagnetizing factor, N∗. Experiment andmicromagnetic simulations give a demagnetizing factor that increases loga-rithmically with the grain size, D. Using a simple analytic model for Hc(D)we show that the key factor that causes the grain size dependence of the co-ercive field is the local demagnetizing field in the region where magnetizationreversal starts.

Several empirical relations between the grain size, D, and the coercivefield, Hc, have been proposed. Nucleation of reversed domains will be initi-ated near surface defects, which may be regions with low or zero magneto-crystalline anisotropy. Smaller grains have lower probability for surface de-fects. This argument was used to interpret the grain size dependence ofthe coercive field in sintered Nd2Fe14B magnets.[5] With the same reason-ing the size dependence of the coercive field was modelled by fitting thecoercive field, Hc, to the logarithm of the surface area, S. With S be-ing proportional to D2, this gives the relationship[6] Hc = a − b ln(D2) orHc = a − b ln(D), using b = 2b where a and b are fitting parameters foundusing the experimental data.

This model was applied by Li and co-workers[7] to fit the size depen-dence of the coercive field in sintered Nd2Fe14B magnets in the range fromD = 4.5 µm to D = 7.5 µm. Alternatively, power laws of the formHc = cD−d, where c and d are also fitting parameters found using theexperimental data, were introduced to describe the grain size dependenceof coercivity. Such a power law was used to describe the coercive field ofsintered magnets with D ranging from 3.5 µm to 7.5 µm.[5] Weizhong andco-workers[8] introduced the model Hc = Ne(αHA−NeffMs). This is a modi-fied version of equation (1) with the parameters α = Neα and Neff = NeNeff ,which are now correlated. The factor Ne was found to be inversely propor-tional to the grain size. This again leads to a power law for Hc(D) with theprefactor c = αHA − NeffMs.

Gronefeld and Kronmuller[9] computed the demagnetizing field near the

3

edge of a Nd2Fe14B permanent magnet. From the local demagnetizing fieldthey derived a local demagnetizing factor which they related to Neff in equa-tion (1). The transverse component of the demagnetizing field increases withdecreasing distance towards the edge and becomes singular at the edge. Nev-ertheless, the magnetization near edges is found to be smooth as shown byanalytic micromagnetic calculations.[10] Similarly, numerical micromagneticsimulations show that the torque from the diverging demagnetizing field isbalanced by the torque from the diverging exchange field near the edge.[11]At a point with a fixed distance from the edge the transverse component ofthe demagnetizing field increases logarithmically with the size of the cube.

Micromagnetic computations of the grain size dependence of coercivityhave been previously reported by various authors. Schabes and Betram[12]clearly show how the non-uniform demagnetizing field in a ferromagneticcube causes the magnetization to rotate out of the uniaxial anisotropy di-rection near the edges and corners forming the flower state. Their studywas for particulate recording. The particles were in the size range from 5nm to 55 nm. The moderate magnetocrystalline anisotropy of particulatemedia is much smaller than that of high performance permanent magnets.Schmidts and Kronmuller[13] computed the coercive field of hard magneticparallelepipeds using the intrinsic magnetic properties of Nd2Fe14B. Theirsimulations were two-dimensional. They assumed translational symmetry ina single direction perpendicular to the easy axis. They included experimen-tal data on the grain size dependence for melt-spun magnets and sinteredmagnets. However, with grain size several other parameters change so thata clear comparison of experiment with simulation was not possible. Forsmall particle sizes with D < 100 nm the numerically calculated coercivefield decreases with a power law, whereas for D > 1000 nm the numericallycalculated coercive field decays logarithmically. Thielsch and co-workers[14]showed that the local demagnetizing field of Nd2Fe14B grains, which havesize and shape typically found in a hot-deformed magnet, causes a shapedependence of the coercive field. The total magnetic field, which is the sumof the external field and the demagnetizing field, initiates the nucleation of areversed domain at the center of an edge. Near the edge where the nucleationstarts the transverse component of the demagnetizing field means there is afinite angle between the total field and the easy axis even when the externalfield is applied at zero angle. The coercive field decreases with the angle ofthe total field according to the Stoner-Wohlfarth angular dependence. Theauthors concludes that the local demagnetizing field at this very point in themagnet significantly influences the size dependence and shape dependence ofcoercivity. Sepehri-Amin and co-workers [15] used micromagnetics simula-tions of poly-crystalline model magnets to explain the grain size dependenceof coercivity in anisotropic Nd-Fe-B sintered magnets. In particular, theyshowed that the magneto-static interaction fields of reversed surface grainsincrease with increasing grain size.

4

In this work we focus on magnetization reversal of a single grain. Weshow that the logarithmic decay of the coercive field with grain size canbe understood by the local self-demagnetizing field of a single grain. Thisfield is non-uniform and initiates the nucleation of reversed domains nearthe edge of the grain. Thus the local magnetostatic field dominates over themacroscopic shape effect.

2 Methodology

We use three-dimensional finite element simulations[16] to compare micro-magnetic theory with experiments. We include melt-spun magnets, (Nd,Cu)infiltrated magnets and sintered magnets in our study, covering a wide rangeof grain sizes (55 nm to 8300 nm). (Nd,Cu) infiltrated magnets have an in-termediate grain size and the grains are almost perfectly isolated by a grainboundary phase. The composition, average grain size, coercive field and αand Neff parameters at room temperature of the Dy-free magnets used inour study are given in table 1. The grain sizes were measured with ScanningElectron Microcopy (SEM) or Transmission Electron Microscopy (TEM) us-ing the line intercept method.[17] The lines to estimate the average grainsize were drawn perpendicular to the direction of the magnetic measure-ment. The measured mean grain diameter is smaller than the actual grainsize because lines do not necessarily intersect at the center of the grain,therefore we correct the grain size with a compensation formula[18, 19].

Sample 1 is a melt-spun ribbon with NdCu infiltration. The NdCu alloywith eutectic composition was infiltrated in the rapidly quenched ribbon at580◦C for 60 min. The weight ratio of NdCu was 40%. Sample 2 is anas melt-spun ribbon with high Nd, B concentration. The wheel speed was20 m/s. Sample 3 is an isotropic sintered magnet produced from a rapidlyquenched ribbon at 650◦C and a pressure of 400 MPa. The NdCu alloy witheutectic composition was infiltrated at 580◦C for 180 min. Sample 4 wasproduced from hot pressing of melt-spun ribbons at 650◦C and a pressure of400 MPa. Die-upsetting was carried out with a strain rate of 1 s−1 at 780◦C.Furthermore, the NdCu alloy with eutectic composition was infiltrated intothe hot deformed magnet at 580◦C for 180 min. The weight ratio of NdCuwas 20%. Sample 5 was produced from hot pressing of melt-spun ribbonsat 650◦C at a pressure of 400 MPa. Die-upsetting was carried out with aslow strain rate of 0.01 s−1 at 780◦C. Furthermore, the NdCu alloy witheutectic composition was infiltrated into the hot deformed magnet at 580◦Cfor 180 min. The weight ratio of NdCu was 40%. Sample 6 is a commercialDy-free sintered magnet. Samples 1 to 3 are isotropic magnets and samples4 to 6 are aligned magnets.

Using an energy minimization method[16] we computed the coercive fieldas a function of particle size for three different shapes: The sphere, the do-

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Table 1: Magnets used for analyzing the grain size dependence of coercivity.

composition (at %) D(nm) µ0Hc(T) α Neff

1 Nd13.9Fe75.5Co4.5B5.5Ga0.6 55 2.66 0.40 0.202 Nd15Fe70B14Ga1 60 2.83 0.42 0.253 Nd13.9Fe75.5Co4.5B5.5Ga0.6 88 2.61 0.43 0.384 Nd13.9Fe75.5Co4.5B5.5Ga0.6 346 2.41 0.48 0.765 Nd13.9Fe75.5Co4.5B5.5Ga0.6 789 1.95 0.45 0.916 Nd14.6Fe76.9Co1.8B6.1Al0.5Cu0.1 8326 1.40 0.45 1.22

decahedron and the cube, which are all perfect particles without any defects.The intrinsic material properties are kept uniform within each particle. Wetake the following intrinsic magnetic properties for Nd2Fe14B: Magnetocrys-talline anisotropy constant K1 = 4.9 MJ/m3, spontaneous magnetizationµ0Ms = 1.61 T, and exchange constant A = 8 pJ/m. The Bloch wall width,δB = π

√A/K1, is 4 nm and the exchange length, Lex =

√A/(µ0Ms), is

1.97 nm. We use a geometrically scaled tetrahedral mesh which is refinedtowards the edges of the magnet. Following Rave and co-workers[11] themesh size along the edges was set to Lex/2.

The micromagnetic results will be compared with a simple analyticmodel that applies the Stoner-Wohlfarth theory locally in the region wheremagnetization reversal is initiated.[14] We evaluate the demagnetization fieldof a uniformly magnetized cube with edge length D using the equation givenby Akoun and Yonnet[20]. We give these equations in the appendix.

3 Results and discussion

Figure 1 shows a plot of the normalized coercive field Hc/Ms versus HA/Ms

evaluated at different temperatures. The data can be fitted to a straight line.According to equation (1) the slope gives the microstructural parameter αand the intercept with the y axis gives the effective demagnetization factorNeff . The temperature dependent values for HA and Ms were taken fromGrossinger and co-workers[21] and Hock[22], respectively. The lines in figure1 are almost parallel to each other which indicates that the α values of themagnets are similar. The sequence of lines from top to bottom starts withthe melt-spun ribbons followed by the infiltrated hot-deformed magnets andthe Dy free sintered magnet at the bottom. This reflects the increase of theNeff with increasing grain size. The microstructural parameters calculatedby fitting the experimental data are listed in table 1.

Figure 2 compares the computed coercive field for a Nd2Fe14B sphereand a Nd2Fe14B cube. For the sphere, the micromagnetics results follow thetheoretical nucleation field for uniform rotation at small diameters (D < Dc)and for the curling mode at larger diameters (D > Dc). The critical diameter

6

Figure 1: Fit of the temperature dependent coercive field, in order to esti-mate the microstructural parameters for the magnets listed in table 1 .

Dc is 10.198 Lex. It is interesting to note that the coercive field of a spherereaches a finite value Hc = 2K1/(µ0Ms)−(1/3)Ms for D � Lex. In contrast,the computed coercive field of cubes decreases with increasing grain sizefor all D > Lex. The deviation from the theoretical prediction has to beattributed to the non-uniform demagnetizing field.

Figure 3 shows the grain size dependence of the coercive field computedfor the cube and the dodecahedron. In addition the plot contains the exper-imental data of the magnets in table 1 together with various other coercivityvalues for sintered magnets taken from literature[5, 6, 25]. The α values ofthe sintered magnets are expected to be comparable to those of hot-deformedNdCu infiltrated magnets. The dashed line in figure 3 is a logarithmic fit toall experimental values. This result confirms the logarithmic decay of thecoercive field with grain size, which is shown here for a wide range span-ning several orders of magnitude. Similarly, the computed values for Hc(D)decay logarithmically (dotted lines in figure 3). The switching field of thecube is lower than that of the dodecahedron. This reflects the difference inthe demagnetizing field near the edges. At the same distance from the edgethe demagnetizing field in the cube is higher than the demagnetizing fieldin the dodecahedron. The computed values for the coercive field are largerthan the experimental values by more than a factor of two. This differencemay be attributed to local defects in the magneto-crystalline anisotropy[4]or soft ferromagnetic grain boundary phases[3] not taken into account in

7

Figure 2: Computed nucleation field of a Nd2Fe14B sphere (circles) and aNd2Fe14B cube (squares) as function of size. Solid line: Theoretical nucle-ation field of a sphere[23]. Dashed line: Theoretical nucleation field of acube[24]. The x-axis gives the diameter of the sphere or the edge length ofthe cube.

the micromagnetic simulations. Further, thermal fluctuations, which arenot taken into account in these micromagnetic simulations, contribute tothe offset between the calculated and experimental coercivities. Thermalfluctuations may help the system to overcome a finite energy barrier withinthe measurement time. The reduction of Hc by thermal fluctuations can beestimated by computing the size of the energy barrier to reversal as a func-tion of the applied field in order to estimate the field required to reduce theenergy barrier to a height of 25 kBT . This method was successfully appliedto compute thermally induced vortex nucleations in permalloy elements [26]and the temperature dependence of coercivity in magnetic recording media[27]. In permanent magnetic grains the reduction in Hc owing to thermaljumps over energy barries is typically estimated at about 20 percent.[28]

Next we analyze the results using the following equation, which wassuggested by Kronmuller and Fahnle[29],

Hc = α∗2K1

µ0Ms− n ln

(D

δB

)Ms. (2)

where α∗ is an effective microstructural parameter. Kronmuller andFahnle showed that the nucleation field of free NdFeB particles follows thelogarithmic law (Equation 2). They also reported that the effective de-magnetization factor Neff = n ln(D/δB) from micromagnetics simulations

8

Figure 3: Reversal field as function of grain size. Circles: Numerical re-sults for the dodecahedron. Full Squares: Numerical results for the cube,perfectly aligned. Open Squares: Numerical results for the cube, field angleof 8 degrees. Diamonds: Experimental data for magnets of table 1. Trian-gles: Experimental data by Ramesh and co-workers[6]. Stars: Fine grainedsintered magnets by Fukada and co-workers[25], Xs: Sintered magnets byUestuener and co-workers[5].

agrees well with experimental results. The finding by Kronmuller and Fahnleclearly shows that the grain size dependence in nucleation-controlled perma-nent magnets is a magnetostatic effect, which leads to a logarithmic decayof coercivity with grain size. The logarithmic law (Equation 2) has to bedistinguished from the coercivity resulting from statistical pinning theory,which would lead to a coercivity proportional to

√lnD.[29] We believe that

the magnets from which we obtained the experimental data for compari-son in this paper are nucleation controlled. The α values (Table 1) are allgreater than or equal to 0.4 which, according to Kronmuller and co-workers[4], strongly indicates that pinning plays no role in the coercivity mechanism.Therefore we restrict our discussion to a single-grain defect-free magnet. In-terestingly, this is sufficient to explain the experimentally-found logarithmicdecay of coercivity. From the slope, n, of the curve Hc/Ms versus ln(D/δB)we can derive a size dependent demagnetizing factor

N∗ = n ln

(D

δB

). (3)

The demagnetizing factor increases with the grain size. The factor n in

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Figure 4: Size dependent demagnetization factor, N∗, computed for a do-decahedron (circles) and the cube (full squares: perfectly aligned, opensquares: 8 degree field angle). Experimentally measured effective demag-netizing factor Neff for the aligned magnets (diamonds) and the isotropicmagnets (triangles) of table 1. The arrows indicate the change of slopecaused by the differences in shape and alignment of the particles.

Equation (3) is related to the slope of the Hc(lnD) curve, which is givenby −nMs. The numerical results (see Figure 4) suggest that n dependson the particle shape and the degree of alignment. The values of n are0.32, 0.27, and 0.14 for the aligned dodecahedron, the aligned cube, and thecube rotated by 8 degrees with respect to the external field, respectively.Using the room temperature values for the magnets in table 1 we obtainn = 0.17. Fitting all experimental values (our own data from table 1 andthe literature values[5, 6, 25]) gives n = 0.18. In addition, Figure 4 givesthe effective demagnetizing factor of the aligned magnets (samples 3 to 6)of table 1. The plot clearly shows that the effective demagnetizing factorderived using Equation (1) increases logarithmically with the grain size.Figure 4 also shows that the demagnetizing factor decreases if the grain ismisaligned. Nevertheless the coercive field of the misaligned grain is smallerthan that of the perfectly aligned sample. This indicates that the first termof Equation (1) and the reduction of α with field angle is dominating. Weconclude that the experimentally found increase of the coercive field withlarger field angle has to be attributed to surface defects [28]. Such defectsare not considered in this work where we restrict our numerical models onlocal demagnetizing effects of perfect particles.

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Figure 5: Demagnetizing field of a uniformly magnetized cube evaluated at adistance of d = 1.2Lex from the edge. Solid line: Component perpendicularto the easy axis. Dashed line: Component parallel to the easy axis.

In the following we develop a simple model that explains the logarithmicdecay of the coercive field with grain size and the slope of Hc(lnD). Inparticular we show that the demagnetizing field at the point where magne-tization reversal starts is decisive for the grain size dependence of Hc.

For small cubes the drop of the coercive field as the field angle is changedfrom zero to eight degrees can be understood by the Stoner-Wohlfarth[30]theory. For a small sphere which switches by uniform rotation the switchingfield is

Hsw = fHA (4)

f =(

cos(ψ)2/3 + sin(ψ)2/3)−3/2

(5)

where ψ is the field angle. For ψ = 8 degrees the reduction factor, f , is about0.7. This partially explains the drop of Hc between the perfectly alignedcube and the cube rotated by 8 degrees for small grain sizes. However thenumerical simulation clearly show magnetization reversal by the nucleationand expansion of a reversed domain so that the Stoner-Wohlfarth theory isnot directly applicable.

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Following the arguments of Thielsch and co-workers[14] we look at theangle of the total internal field in the region where magnetization reversalstarts. They showed that Equation (4) can be applied locally, in order toestimate the coercive field, whereby ψ is to be replaced by the angle of thetotal internal field, ψt, at the point where nucleation reversal starts. Theyfound that in Nd2Fe14B particles with a rectangular prism shape magneti-zation reversal starts at the center of an edge. Next we apply the Stoner-Wohlfarth theory locally within the reversal volume near the edge of thecube.

We use an analytic expression[20] for the demagnetizing field of a cubeand evaluate the demagnetizing field at a distance d = 1.2Lex from the centerof an edge. Figure 5 shows the components of Hd parallel and perpendicularto the easy axis. With increasing size of the cube the perpendicular compo-nent of the demagnetizing field increases. This in turn leads to an increaseof the angle ψt. The external field is applied either parallel to the easy axisor at an angle of ψext = 8 degrees. The components of the external field,Hext, are

Hext,‖ = −Hext cos(ψext) (6)

Hext,⊥ = Hext sin(ψext). (7)

The components of the exchange field, Hx, in the reversal region are evalu-ated as

Hx,‖ = 0, (8)

Hx,⊥ =1

µ0Ms

A

d2. (9)

The total internal field Ht is the sum of the the demagnetizing field, theapplied external field, and the exchange field:

Ht = Hd + Hext + Hx. (10)

With the simple model we compute the coercive field as follows: We suc-cessively increase Hext. For each value of Hext we evaluate Ht and computeψt. We denote the value of Hext when

Hext ≥ Hsw(ψt) (11)

the approximate coercive field Hc. Figure 6 shows that Hc coincides with themicromagnetically computed coercive field. In order to obtain a quantitativematch between the simple model and the micromagnetic result, we have toinclude the exchange field, Equations (8) and (9), in the simple model. Thisis different from the qualitative treatment by Thielsch and co-workers[14]who only considered the external field and the demagnetizing field. In thesimple model the only input that changes with the grain size is the perpen-dicular component of the demagnetizing field. With increasing grain size

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Figure 6: Computed coercive field as a function of the grain size. Com-parison of the full micromagnetic results (squares) with a simple model(triangles) for the coercive field. Full symbols: Perfectly aligned cube. Opensymbols: Cube, field angle of 8 degrees.

Hd,⊥ increases. As a consequence the angle of the total internal field withrespect to the easy axis increases. This is clearly seen in Figure 7, whichshows ψt evaluated at Hext = Hc for different grain sizes. For the perfectlyaligned case the angles are between 4.5 degrees and 16 degrees. This rangeis shifted towards higher angles (14 degrees to 27 degrees) for the cube with8 degrees misalignment. This difference explains the reduction of the slopeof Hc(lnD) in the case of a misalignment of 8 degrees. At small angles thefactor f , see Equation (5), changes rapidly with ψ. For larger angles f(ψ)becomes flatter.[30]

The above model shows that the total field in the region where mag-netization reversal starts determines the coercive field. In particular theangle of the total internal field with respect to the easy axis in the nucle-ation region, ψt, is important. Due to their influence on the perpendicularcomponent of the demagnetizing field a change in grain size or grain shapewill change this angle. Clearly, the alignment of the grains will influence ψt.Other microstructural effects that will have an influence at the local reversalconditions are the nature of the grain boundary phase[3] and soft magneticdefects. Grain boundary phases may change the local exchange field andin turn alter ψt. Defects in the local magneto-crystalline anisotropy willchange HA in the nucleation region and thus modify Hsw. In a real magnet

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Figure 7: Angle of the total internal field, ψt, with respect to the easy axisat a distance of d = 1.2Lex from the edge. Full squares: Perfectly alignedcube. Open squares: Cube, 8 degrees rotated.

the interplay of various microstructural effects will determine the slope ofHc(lnD).

4 Conclusion

We showed that the numerically computed coercive field as a function ofgrain size for a single Nd-Fe-B grain can be fitted perfectly to a logarithmiclaw, Equation (2). In the simulations we assumed the most simple model toexplain this effect, a single isolated particle without defects. In our modelthe logarithmic decay of the coercive field can be attributed neither to thedensity of surface defects nor to stochastic domain wall pinning. Thereforewe conclude that the logarithmic decay in polyhedral grains results frommagnetostatic edge effects. Indeed, the equations for the demagnetizingfield show a logarithmic singularity near the edge of a grain (see Appendix).

In summary, we confirmed the logarithmic decay of Hc with grain size fora simple micromagnetic model and a wide range of Nd2Fe14B magnets. Theresults suggest that the logarithmic decay of coercive field with increasinggrain size results from the logarithmic increase of the demagnetizing fieldnear the edges of a grain. At this very location the torque exerted by the lo-cal field onto the magnetization initiates the formation of a reversed domain.With increasing particle size the torque that rotates the magnetization outof the anisotropy direction becomes larger and domain formation happens at

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lower external fields. The slope of Hc(lnD) depends on the microstructuralfeatures including the grain shape, the degree of alignment and most likelythe nature of the grain boundary phases.

Acknowledgments

This work is based on results obtained from the future pioneering program“Development of magnetic material technology for high-efficiency motors”commissioned by the New Energy and Industrial Technology DevelopmentOrganization (NEDO). We acknowledge the financial support from the Aus-trian Science Fund (F4112-N13).

Appendix

The components of the demagnetizing field of a uniformly magnetized par-ticle of cuboidal shape and dimensions 2a, 2b, and 2c which is magnetizedalong the z axis are:

Hd,‖(x, y, z) =Ms

4π

1∑i=0

1∑j=0

1∑k=0

(−1)i+j+kA(ξi, ηj , ζk) (12)

Hd,⊥(x, y, z) =Ms

4π

1∑i=0

1∑j=0

1∑k=0

(−1)i+j+k+1Lη(ξi, ηj , ζk) (13)

whereξi = x− (−1)ia (14)

ηj = y − (−1)jb (15)

ζk = z − (−1)kc (16)

and

ρ(ξ, η, z) = (ξ2 + η2 + z2)1/2 (17)

A(ξ, η, ζ) = arctan

(ηξ

ζρ

)(18)

Lη(ξ, η, ζ) = log(η + ρ) (19)

Here Hd,‖ and Hd,⊥ are the components of the field parallel and perpen-dicular to the magnetization. The origin of the coordinate system is at thecenter of the cube. Fig. 5 gives the field close to the edge of a cube withedge length D = 2a = 2b = 2c.

15

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