Journal of Magnetism and Magnetic Materials 86 (1990) 177-183 177 North-Holland

INFLUENCE OF GRAIN ORIENTATION ON THE COERCIVE FIELD IN Fe -Nd-B PERMANENT MAGNETS

G. MARTINEK and H. KRONMULLER Max-Planck-lnstitut f~r Metallforschung, Institut ffir Physik, Stuttgart, Fed. Rep. Germany

Received 15 September 1989

The nucleation field for homogeneous rotation is calculated numerically, including higher order anisotropy constants. The results are combined with a Gaussian distribution of grain orientations in order to discuss the dependence of coercivity on the degree of grain alignment. For the degree of alignment found in sintered magnets, the coercive field is seen to be reduced severely, its value being closer to the minimllm nucleation field for misaligned grains than to the theoretical value for ideal orientation. An analysis of the experimental coercive field of sintered F e - N d - B magnets shows that the coercive field is determined by the minimum nucleation field for misaligned grains. An expression for this field is given, which for small values of K 2 takes the simple form H~nff i(K1 + K2)/J s.

1. InO'oduction

The smallest theoretical coercive field of ideally oriented ellipsoidal particles was shown by Brown [1] to be

2K1 - - NuM , (1)

(K 1 -- anisotropy constant, NI~-- demagnetization factor parallel to the easy axis, J~ffi#0M, ffi spontaneous polarization and magnetiTation).

Experimental values of the coercive field are generally considerably smaller than this result. This discrepancy is usually attributed to the mi- crostructure of real permanent magnets, namely sharp edges, comers, etc., which cause strong de- magnetizing fields [2,5], or regions with disturbed material parameters (e.g. at grain boundaries) [8] where nucleation is facilitated.

Another factor reducing the nucleation field is the grain orientation. As shown by Stoner and Wohlfarth [3], the theoretical coercive field de- creases for oblique applied fields, leading to a reduction of coercivity to about half the value given by eq. (1) for randomly oriented particles.

In the case of Fe14Nd2B, the angular depen- dence of the nucleation field as well as its temper-

ature dependence are modified because the de- scription of the magnetocrystalline anisotropy re- quires the use of higher order anisotropy constants [9,11,12].

Thus, in order to discuss the experimental coercive fields measured in F e - N d - B permanent magnets, the nucleation field for arbitrarily ori- ented grains, including higher order anisotropy constants, has to be calculated. In a second step, these results will be combined with the distribu- tion of grain orientations of aligned magnets to calculate the angular and temperature dependence of the coercive field.

2. Nucleation fields

2.1. Gibbs free energy

In the following, a uniaxial single crystalline particle in an applied field H "will be considered. The total energy-density Ot comprises four contri- butions

¢, ffi ¢o, + eK + + ¢ . , (2)

which depend on the direction of the spontaneous magnetization M, as given by the angle ~p with

0304-8853/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

178 G. Martinek, H. Kronmfiller / Grain orientation influence in F e - N d - B magnets

respect to the c-axis which is taken as the z-axis of the one-dimensional problem.

1. Exchange energy

• ox =A -d7 / (3)

(A = exchange constan0. 2. Crystal anisotropy energy (neglecting

planar anisotropy)

~i~ = K1 sin2q ~ + K2 sin4q ~ + K3 sin6~ + " " - (4)

3. Strayfield energy (for ellipsoidal particles with uniform magnetization and a c-axis parallel to the rotation axis of the ellipsoid)

• s = ½ oM/(N,, cos2 + N . sin ). ( 5 )

4. Magnetostatic energy

#H = - Hext J~ cos(q0 - tk) (6)

(t/, = angle between the applied field and the c-di- rection).

2.2. Nucleation fields

The equilibrium states of the particle are given by the condition that 0 t be an extremum, which for homogeneous magnetization can be written as

do t d ~ = 0 . (7)

Using eqs. (3)-(6), a relation between the applied

field and the magnetization direction is obtained

H ( ~ ) = - { ( 2 r ; +4K2 sin~, + 6K3 sin4,)

xs in cp cos ~}{Js s in(~0-~k)}- ' . (8)

Here K 1' = K 1 + K d includes an effective ani- sotropy constant describing the effect of the de- magnetizing~fields:

Kd= ½PoM/(N± -Nil ). (9)

The following calculations will be done for a spherical particle ( K d = 0).

Eq. (8) describes both stable and unstable equi- librium states. The stability is determined by the sign of the second derivative of the Gibbs free energy or, alternatively, the first derivative of eq. (8), which, following Brown [4], can be shown to be connected by the simple relation

OH 1 a2~t - - = . (10) a~ Js sin(cp - ~ ) acp 2

The behavior of the particle can thus be com- pletely determined from the H ( ~ ) relation (8). This is shown in fig. 1, where the saturated par- t ide is subjected to an external field with the angle ~k0 = ~r-~k to the negative c-axis. On increasing the field, the magnetization first rotates reversibly out of the c-direction until at the nucleation field H N a spontaneous rotation to a new stable equi- librium ~0' occurs. The nucleation field is given by

re~.

ir

H~t

H tk~Vm]

4 0 0 0

~ 2000

0

-2000

/ I \

I

I I I I I I t I ~1 I I I I I i

50 100 150 ~[o]

I

Fig. 1. Behavior of a particle in a field appfied at the angle ~0 to the negative c-axis. The fat lines in the H(cp) diagram [eq. (8)] show stable states, the dashed line unstable ones. The material parameters of FeliNd2B at room temperature [11] are used.

G. Martinek, H. Kronmigler / Grain orientation influence in Fe-Nd-B magnets 179

H N

I'( 1 = - K ~

/ 0 ~ 90

Fig. 2. Angular dependenca of the nucleation field as calcu- lat~ numerically (solid lines) and as given by eq. (12) (dash~

lines). The ratio K1/K 2 is varied while K 2 is kept constant.

the maximum of the H(cp ) curves shown in eq. (8), where the second derivative of • changes its sign (Ot '= 0). HN(~0) is shown in fig. 2 for different ratios K 1 / K 2. For elevated temperatures, where higher anisotropy constants are negligible, it is given by the expression derived by Stoner and Wohlfarth [3]

2K1 1 (11) H s ( ~ ° ) = d~cos ¢o [1 + tan t~2/3] 3/2"

At lower temperatures, the nucleation fields are substantially affected by the higher order ani- sotropy constants. A modification of eq. (11) con- taining K 2 was derived in ref. [7] by means of a linear perturbation calculation

2K 2 (tan i]/O) 2/3 ] UN(tpo) =HfN°)(tpo) 1+ ~ l + ~ - a n ~ / 3 "

(12)

Here H~ ) is the unperturbed solution of eq. (11). Plots of this expression are also included in fig. 2. A comparison of eq. (12) with the numerical calcu- lations shows a very good agreement for K 2 < ~K v For larger values of K 2 or nonvanishing ani- sotropy constants of still higher order the numeri- cal results have to be used.

An expression for the minimum of the Hs(6o) curves can be found by equating the maximum slope of • K+ • s and the minimum slope of 0H(0~[ ran= -HextJs) . In this case, both condi- tions 0 t = 0 and Ot '= 0 are obviously fulfilled. The minimum nucleation field including K 2 is thus found to be (for K 2 > 0, K i > -2K2)

Js W - + 3

x W ~ +1 - - +3 ,

with

(13)

W = ~-~2+1 + 8 .

The minimum of H N is found at an angle of the applied field given by

= 9 0 ° - a r c s i n 3 - + w .

( 1 4 )

An approximate expression for H~ in may be de- rived from an expansion of eq. (13) for small K 2

KI+K 1], 0 < K 2 < ½K 1. (15)

3. Influence of grain alignment

In order to compare the theoretical coercive fields for single particles with experimental values of coercivity in permanent magnets, the distribu- tion of grain orientations has to be taken into account [7,14,16]. The procedure has been de- scribed in ref. [7]. It turns out that, for the degree of aIi t, nment that is usually found in sintered magnets [6,13,15], the angular dependence of coercivity for single grains is flattened si~,nlfi- cantly. The calculated coercive field is thus almost

180 G Martinek, H. Kronmi~ller / Grain orientation influence in Fe-Nd-B magnets

~oH It]

® 2 'N(O)~,

! 0

S ® ®

L I

~'190 Fig. 3. Angular dependence of coercivity mHc (dashed line) as calculated from the nucleation field for single particles HN (if0) (solid line). The calculation of the magnetization curves is shown schematically: N(O) is the number of grains at angle 0 to the applied field. The shaded fraction of grains has reversed its magnetization at the given field He= t which is applied at the

angle ~b~ to the ali tmment direction.

independent of the angle of the applied field, its value being close to the minimum of the HN0k0) curves. This is demonstrated in fig. 3, where a distribution of grain orientations

P( 0 ) = Ne -(° /°°) ' , ( 1 6 )

with 00 = 20 °, is combined with the nucleation field curve Hr,(fro) for Fe14Nd2B at room tem- perature for the calculation of the magnetization curves. Here, N is given by the condition

fo~'/2P(O) sin 0 dO -- 1. (17)

The coercive field shown is defined as the field of maximum susceptibility in the demagnetization curves and denoted by mHc.

Fig. 4 shows the calculated coercive field jHc as a function of 00. In addition, the ratio of the remanence values for saturation parallel and per- pendieular to the al i~ment direction, B~./B~, is shown. This remanence ratio is often used as a fast measure of the degree of grain ali~mment.

The experimental angular dependence of the coercivity for two sintered magnets with the same remanence ratio, B~t/B~ffi4.3, as measured at room temperature, is shown in fig. 5. A sample with the composition FeTsNdls.sB~.s and p0yHc =

~(0") E ............................................ 20

=_ _._,__~ ............................. __--_ 10

............ ~ 5

. . . . , . . . . . . . . . t . . . . . . . . .

10 15 20 25 30 0 Oo

Fig. 4. The coercive field j H c (solid line) and the remanence ratio B~t/B~ (dashed line) as a function of the texture param- eter 0o. The nucleation fields for ideally oriented grains HN (0 ° ) and the minimum nucleation field H ~ m for misaligned

grains are also indicated.

1.1 T is compared with a sample where 1.5 wt% Al203 was added during milling [17], leading to a coercive field of pojHc = 1.74 T. Although the intrinsic material parameters of the two samples do not differ significantly [10], their angular de- pendencies of coercivity are clearly different. The specimen with the higher coercive fields reveals a less significant angular dependence. Nevertheless, a comparison with the calculated angular depend- ency for independent grains (fig. 3) shows no satisfactory agreement. In particular, no minimum for oblique applied fields is observed in the ex=

2.0

mH H "0°'= O1.1T

1.5

I.O

.5 t I I I I I I I I 0 30 60 ~. 90

Fig. 5. Angular dependence of coercivity mHc for two F e - N d - B sintered magnets. The fields are normalized to the values for

,,t, o = 0.

G. Martinek, H. Kronmigler / Grain orientation influence in Fe-Nd-B magnets 181

perimental curves. This suggests that the coercive field is given by the minimum coercive field for misaligned grains even when the applied field is parallel to the alignment direction. Possible ex- planatious for this behavior will be discussed in section 5.

4. Temperature dependence of the coercive field

The coercive field of sintered magnets is usually described by the relation [14,18,19]

He = aHiN d~ -- NenM s. (18)

Here Nef f is an average effective demagnetization factor describing the internal stray fields acting on the grains and H ~ ca1 is the theoretical nucleation field for perfectly oriented (~k0 ffi 0) particles (HiNd ~f f i 2 K I / J s for vanishing higher order ani- sotropy constants). The factor a, according to

a = aKa~, (19)

describes the reduction of the nucleation field due to inhomogeneous material parameters at grain boundaries (aK) [8] and the influence of grain misorientation (a~), i.e.

H c = ( X K ~ t @ H i N deal - - N e n M s. (20)

In principle, ot~ is equal to the calculated ratio He/HN(~0 = 0) as given in fig. 4. For the degree of alignment found in sintered magnets, H e is seen to be reduced to a value close to the mini- mum nucleation field for misaligned grains. The absence of a minimum in the experimentally ob- served angular dependence of coercivity suggests that it is in fact reduced all the way to H ~ . The nucleation fields HiNd ~ and H~ nin for Fel4Nd2B are shown in fig. 6.

Rewriting eq. (20) by replacing HiN d~a by H N ffi a~,HiN d ~ gives

He HN "~s = aK-~'s -- Neff; (21)

a linear relationship between H c , / M s and H N / M s is found (the temperature dependence of a K can be neglected in the case of Fe14Nd2B ). In fig. 7 this relation is plotted by using experimental

I-IN [kA/m]

I0000

5000

0 I I I I I

I00 200 300 400 500

T [K] Fig. 6. The calculated nucleation fields for Fe14Nd2B for particle orientation parallel to the applied field and the mini-

mum nucleation field for misaligned particles.

coercive fields of a sintered magnet with composi- tion Fe7.sNdls.SB6.5. For HN, both H N (•0 = 0) and H ~ n are used. A linear relationship is only found if

a,~H~ ~ = H ~ i" (22)

is used in eq. (20). From this plot a K = 0.71 and Nef f - - 1.1 are derived. The linearity does not hold for temperatures below 170 K. This is probably due to the negiection o f in-plane anisotropy in the determination of the anisotropy constants as well as in the calculation of the nucleation fields.

1,5

HC/M o S

,o 0 1.0

El

. 5 m In r,I

O. , , , , , , , , i . . . . . , I , , , , , , , , ,

0 1 2 .3 4

HN/Ms

Fig. 7. Test of relation (21) for H N = H~ a~ (O) end for HN = H~" (<>).

182 G. Martinek, H. Kronmiiller / Grain orientation influence in Fe -Nd-B magnets

5. Discussion and condusion

The numerical calculations presented show that, due to imperfect grain alignment, the coercive field in sintered magnets is determined by the minimum nucleation field for misaligned grains rather than by the value for ideal alignment. Fur- thermore, the angular distribution of c-axes leads to a weak dependence of H c on the angle of the applied field (fig. 3).

While the predicted angular dependence of coercivity is found in P r -Fe -B sintered magnets (at lower temperatures) [20] and in hexaferrites [21], a considerable deviation from the theoretical angular dependence has been found in the present measurements on N d - F e - B sintered magnets. Similar deviations have been found by Givord et al. [14]. This deviation can be qualitatively ex- plained by local internal stray fields, which are found to be of the same magnitude as the applied field [14,18,19], however of fluctuating orientation.

1. Strong local stray fields, arising at sharp edges, corners, etc., do in general not act in the direction of the applied field, thus changing the angular dependence of the nucleation field of the individual grains [5].

2. Additional local demagnetizing fields are created near grains with reversed magnetization, leading to a magnetostatic coupling between ad- jacent grains [18,22], which may explain the ab- sence of a minimum in the experimental angular dependence of coercivity. This model in fact pre- dicts a coercive field which is independent of the angle of the applied field. The magnetostatic cou- pling mechanism becomes most effective if the magnetization in each grain is reversed completely after the nucleation of reversed domains. Such a complete reversal by domain wall displacements requires external fields of

H r > N M s (23) coS 1~0 '

where N is the demagnetizing factor of the grain and domain-wall pinning effects are neglected. Since H r becomes rather large for ~b o --* ,a/2, these grains do not reverse their magnetization sponta-

neously, i.e. the magnetostatic coupling with neighboring grains is less effective, thus explaining the increase of coercivity for large angles ~k0 (see fig. 5).

In F e - N d - B magnets, the elaborate determina- tion of a¢ was found to be not necessary to describe the influence of imperfect grain align- ment. Instead, it is sufficient to use [see eq. (20)]

H c = o /H~ nin _ Nef f Ms (24)

for eq. (18). This should be true for all nucleation hardened magnets that do not show a pronounced minimum in the angular dependence of coercivity. For materials with vanishing higher order ani- sotropy constants H ~ = K 1 / J s = ½ H ~ e~ [3] and therefore the temperature dependence of coerciv- ity is not affected by this change. When higher order anisotropy constants cannot be neglected, however, the temperature dependence of H~ ~i~ is different from that of HiNd ~ and expression (13) or the numerically calculated H ~ in have to be used. In the region where K: is small (i.e. above room temperature for Fe14NdzB ), eq. (15), or simply

n ~ nin = K1 + K 2 , 0 < K 2 < ~-K 1, (25)

represents a good approximation.

References

[1] W.F. Brown, Rev. Mod. Phys. 17 (1945) 15. [2] H. Zijlstra, in: Ferromagnetic Materials, vol. 3, ed. E.P.

Wohlfarth (North=Holland, Amsterdam, 1982). [3] E.C. Stoner and E.P. Wohlfarth, Phil. Trans. Roy. Sec.

240 (1948) 599. [4] W.F. Brown, Magnetostatie Principles in Ferromagnetism

(North-Holland, Amsterdam, 1982) [5] M. Gr~Snefeld and H. Kronmiiller, J. Magn. Magn. Mat.

80 (1989) 223. [6] K.-D. Durst and H. Krortmiiller, Prec. 8th Intern.

Workshop on Rare Earth Magnets, ed. K. J. Stmat (Univ. of Dayton, Ohio, 1985) 725.

[7] H. Kronmiiller, K.-D. Durst and G. Martinek, J. Magn. Magn. Mat. 69 (1987) 149.

[8] H. Kronmllller, Prec. 2nd Intern. Symp. Coercivity and Anisotropy of RE-Trans. Met. Alloys (San Diego, 1987) p. 1.

G. Martinek, H. Kronmiiller / Grain orientation influence in Fe-Nd-B magnets 183

[9] K.-D. Durst and H. Kronmllller, J. Magn. Magn. Mat. 59 (1986) 86.

[10] S. Hock and H. Kronmikller, Proc. 9th Intern. Workshop on Rare Earth Magnets, Bad Soden, 1987, eds. C. Herget, H. Kronmfiiler and R. Poerschke (Deutsche Physikalische Gesellschaft) p. 275.

[11] S. Hock, Dissertation, Universitllt Stuttgart, 1988. [12] O. Yamada, H. Tokuhara, F. Ono, M. Sagawa and Y.

Matsuura, J. Magn. Magn. Mat. 54-57 (1986) 575. [13] D. Givord, A. Li6nard, R. Pettier de la B~thie, P. Tenaud

and T. Viadieu, J. Phys. C 6 (1985) 313. [14] D. Givord, P. Tenaud and T. Viadieu, IEEE Trans. Magn.

MAG-24 (1988) 1921. [15] S.R. Trout and C.D. Graham, IEEE Trans. Magn. MAG-

12 (1976).

[16] L. Jahn, R. Schumann and V. Christoph, Phys. Stat. Sol. (a) 88 (1985) 595.

[17] L. Kiss, G. Martinek, A. Forkl and H. Kronmtiller, Phys. Stat. Sol. (a) 114 (1989) 685.

[18] E. Adler and P. Hamann, Proe. 8th Intern. Workshop on Rare Earth Magnets, University of Dayton, Ohio, 1985, ed. K.J. Strnat, p. 747.

[19] H. Kxonmiiller, K.-D. Durst and M. Sagawa, J. Magn. Magn. Mat. 74 (1988) 291.

[20] G. Martinek, H. K.ronmikller and S. I-Iirosawa, to be pubfished.

[21] D.V. Ratnarn and W.R. Buessem, J. Appl. Phys. 43, (1972) 1291.

[22] J. Pastuschenkow, K.-D. Durst and H. K.ronmiiller, Phys. Stat. Sol. (a) 104 (1987) 487.