ferromagnetic coercivity and applied field orientation

ELSEVIER Journal of Magnetism and Magnetic Materials 147 (1995) 331-340

~ Journal of mnagnetlsm magnetic

~ l ~ materials

Ferromagnetic coercivity and applied field orientation

Brandon Edwards a, D.I. Paul b,, Math Department, Unicersity of British Columbia, Vancouver, BC, Canada

Materials Science Department, MIT, Cambridge, MA 02139, USA

Received 10 October 1994; in revised form 4 January 1995

Abstract

We analyze the relationship between the coercive force of a ferromagnetic material and the angle of the applied magnetic field. The material is assumed to contain a ferromagnetic domain wall as well as a planar defect parallel to the wall and it is further assumed that the dominant mechanism determing the coercivity is that of pinning of the wall by the defect. Our formulation takes into account the spatial dependence of the direction of magnetization along the normal to the plane of the defect. Numerical solutions are obtained for the resulting nonlinear differential equations and analysis is done on the roles of the anisotropic, magnetostatic, and exchange energies in determining the behavior of the coercivity as the direction of the applied magnetic field is varied. Our results show that, in contradiction to previous thought, the inverse cosine of the applied field angle is not a good approximation to the coercivity dependence unless the coercivity is about two orders of magnitude smaller than the anisotropy field. Also, there exist ranges of parameter values for which the domain wall pinning coercivity decreases as the angle between the applied magnetic field and the anisotropy field increases - a behavior previously assumed to occur only when the coercivity is dominated by nucleation rather than pinning of domain walls. Thus, caution must be exercised when using the angular dependence of the applied field to determine the mechanism of magnetic reversal of a given material.

I. Introduction

A central concern in the study of permanent magnets such as NdFeB and SmCo is determining which magnetic reversal mechanism dominates the coercivity. The measured coercivity fields, H e, are generally found to be 1 / 3 to 1 / 1 0 that of the theoretical limit given by H A = 2 K / M , where K and M are the anisotropy and magnetization constants, respectively [1,2].

An important aspect of this problem is the dependence of the coercivity, He, on OH, the angle of the applied magnetic field with respect to the easy direc-

* Corresponding author. Tel: + l (617) 253-3306.

tion of magnetization of the material. This angular dependence has been measured in sintered RFeB magnets by the group at CNRS in Grenoble [3]. They find that at room temperature the increase in He(0 H) with 0 H when Hc(0) = 10 kOe corresponds approximately to a 1/cos(O n) relationship. A theoretical derivation of this result had been carried out in a previous paper [4]. For H e ( 0 ) = 17 kOe, there is a reduced increase in Hc(O H) with O N. Finally, a decrease in Hc(0 H) as 0 H increases occurs at small values of O H in PrFeB magnets. The authors of Ref. [3] state that they are unable to understand the last result on the basis of either coherent rotation of the magnetization or domain wall propagation.

Several limiting behaviors in the coercivity can be considered. When H c is of the order of H A , the

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332 B. Edwards, D.L Paul/Journal of Magnetism and Magnetic Materials 147 (1995) 331-340

Stoner-Wohlfarth model [5] should apply, wherein H c decreases as 0 u increases from 0 ° to 45 ° with H~ returning to its original value as 0 n increases further from 45 ° to 90 °. At the opposite limit, when Hc << H A, arguments have been given for a I / c o s ( O H) or inverse cosine dependence of the coercive field [6].

Results by both Sakuma et al. [7] and KronmiJller et al. [8] indicate that there exist ranges of magnetic parameter values wherein coercivity limited by the nucleation of domain walls shows deviations from an inverse cosine angular dependence, but it is generally accepted that coercivity limited by domain wall pinning follows the inverse cosine relation, thus yield- ing insight into the mechanism of magnetic reversal through analyzing the angular dependence of the coercivity of different materials. We show that this is not so, that there also exist strong deviations from the inverse cosine relation when domain wall pinning is the mechanism limiting magnetization reversal.

It is evident, due to the significant deviation of the measured coercive fields of materials such as N d - F e - B from the theoretical value H A, that the micromagnetic phenomena accompanying magnetization reversal must be included in order to obtain accurate results. One fundamental micromagnetic aspect of domain walls is the existence of nearest- neighbor interactions between the magnetic moments giving a spatial dependence to the magnetic orientation. The formulation of the Stoner-Wohlfarth curve for magnetic reversal by rotation [5] as well as derivations of the i / c o s ( O H ) angular dependence [4,8] neglect this spatial dependence.

In 1990, Sakuma et al. [7] presented theoretical derivations of the coercivity as a function of the angle of the applied field for the case of a domain wall nucleation-dominated coercivity in a material containing a defect. Their methods, first introduced by Friedberg and Paul [9], took into account the spatial dependence of the magnetization.

In this paper we derive the angular dependence of the coercive force for a domain wall pinning dominated coercivity. We also take into account the spatial dependence of the magnetization. We discuss the effects of variations in the magnetic parameters of the defect with respect to the host material in order to gain insight into the physical phenomena accompanying magnetic reversal. We then compare our

HOST ] DEFECT HOST A1 ' H1 ' M1 / A2' K2, M2 A1 ' K1 ' M1

I

I / Fig. 1. Diagram of the one-dimensional magnetic host matrix having magnetic parameters Ai, K1, and M 1 and containing a defect region ( - D / 2 , D/2), characterized by the parameters A2, K2, and M 2. The angles in the y - z plane, O, of a dipole along the array and 0 N, of the applied field H are taken with respect to the y-axes. The anisotropy axes is parallel to the y-axes.

curves with the work mentioned above as well as other work. Among other topics, we examine the qualitative diferences, if any, of an angular dependence curve derived from a nucleation-dominated mechanism as opposed to a pinning-dominated mechanism.

2. Model

We analyze the coercive force of ferromagnets when such force is dominated by domain wall pinning due to impediments in the material. Consider an array of magnetic dipoles representing the host magnetic material containing a defect lying within the region x = - D / 2 to x = D / 2 (see Fig. 1). The host matrix on both sides of the defect has uniform magnetization M t, first-order anisotropy constant K t, and first-order exchange energy constant A~. The impediment has entirely independent uniform magnetic properties K2, M2, and A 2. The easy axes of magnetization are in the y-direction and an external magnetic field, H, is applied at an angle 0 n as measured from the y-axis in the y - z plane. The orientation of a given dipole in the array is taken as the angle 0, also measured from the y-axis in the

B. Edwards, D.1. Paul/Journal of Magnetism and Magnetic Materials 147 (1995) 331-340 333

y - z plane. We assume that 0 has a one dimensional spatial dependence. The orientations of the end dipoles are fixed with 0 = 0 ° at x = - ~ and 0 = 180 ° at x = % signifying the presence of a Bloch wall along the array. We wish to find the value, H c, of the external magnetic field for which the wall can no longer be held by the impediment, and thus magnetic reversal of the entire medium occurs. This model has already been discussed for the case in which the external magnetic field is applied in the same direction as the easy axes of magnetization, i.e. (O n = 0 °) [9,10]. We now extend the model for 0 n ~ 0 ° to determine the change in the coercive force due to a change in the angle at which the external magnetic field is applied with respect to the anisotropy axes.

The energy for the ith region (i = 1, 2), is given by

I K i s in : (O) - HM, cos( 0 - O n )

f

~x [ 1

t dx ] ] dx. (1)

Minimizing the energy in the three regions using the boundary conditions that 0 = 0 ° and 180 ° at x = - ~ and % respectively and that Ai (dO/dx ) is continuous at x = - D / 2 and D / 2 , we get

w=fC°S<°0( . 0 ) - 1 [ s i n 2 ( 0 ) "cos(02) sin

- h F E -1 c o s ( 0 - OH) 71- ? 7 ] - 1 / 2 d cos(0), (2)

where ~ is given by

T/= ( E - ' - 1) sin2(01) - h E - l ( 1 - F )

×COS(01 -- 0n) + h E - l c o s ( O n ) . (3)

The relationship between the quantities 01 and 02 , (the angles giving the orientation of the dipoles with respect to the easy direction of magnetisation at x = - D / 2 and D / 2 , respectively), is given by

( 1 - E -1) s i n e ( 0 2 ) + h E I ( 1 - F ) cos(02 - 0n)

+ hE 1 COS( O H) + 71 : O, ( 4 )

along with the condition cos(01) > cos(02).

Here we have also introduced the dimensionless parameters E, F, h, and W:

A2K 2 A 2 M 2 HAl 1 D

E - A1K1 F A1M1 h = K1 W 6 '

used in Ref. [9], where D is the actual width of the defect and ~ = (A2 /K2) 1/2 is the domain wall half- width within the defect region. We assume that the defect region is less magnetic than the host material and therefore restrict the values of E and F to be less than unity. Analysis similar to that carried out in this paper can be done for materials in which E or F is greater than unity.

Given values for the normalized external field h and O H , there is a set of 01's for which a 02 in (01, 180 °) satisfying Eq. (4) will exist, and for which W(Ol, 02) < % where W(01, 02) is determined by Eq. (2). This gives rise to a set (Wol} of normalized defect widths. These are values of W for which there exists a static domain wall solution that satisfies the given boundary conditions. In particular, 0( - ~) = 0 ° and 0(~) = 180 ° are satisfied and thus a domain wall is still present. Thus for all widths W in {Wol}, the field h is insufficient to drive the wall over the defect.

The minimum, Wc, of the set {W01} corresponds to the critical width at which h is now sufficient to drive the wall over the defect region and magnetic reversal begins. As one would suspect, it was found that W c increases as a function of h. Thus any field smaller than h would be insufficient to drive the wall across a defect of width Wc(h) as contrasted to a field greater than h. Therefore h c = h is the coercive force corresponding to the given defect width We(h).

3. Procedure

In most of our calculations, we assigned a value to W and sought to determine the coercive force for a host matrix containing a defect of this width. This was done by determining the unique h c such that W¢(h c) = W. Solutions to Eq. (2) in this procedure were determined numerically. The integral was ap- proximated using Simpson's rule with adaptive quadrature and an error tolerance of 1%.

Solutions to Eq. (4), which determines the lower limit of the integral in Eq. (2) in terms of the upper

334 B. Edwards, D.I. Paul~Journal of Magnetism and Magnetic Materials 147 (1995) 331-340

limit, were also obtained numerically using an algorithm that searched for appropriate values of cos(02) along the interval [ - 1 , cos(01)].

It is possible to have two values of cos(02) that solve Eq. (4), although as shown below we are only concerned here with the larger value. Indeed, to find W~ we need to determine the minimum W for which h is still insufficient to drive the wall across the defect, that is, the minimum W that Eq. (4) will give for the many cos(01), cos(02) pairs determined by h. The integrand in Eq. (2) is nonnegative, so that if there exists c o s ( 0 ~ ) < cos(02) where both are less than cos(01) and solutions to Eq. (4), then

w'= fc°s<%[cos( 0)] d cos(0) " COS(01 )

> fc°s~<)I[cos(O)] d cos(0) = w, % O S ( 0 1 )

w h e r e / [ c o s ( 0 ) ] is the integrand in Eq. (2). Thus for a given cos (0t) , we take the greatest cos(02). Details of the search algorithm are given in the

Appendix. In Figs. 2 - 7 in which we plot the ratio of normal-

ized coercive forces hc (O, ) /hc (O) versus the angle of the applied field O H , the maximum value of 0/4 taken for our calculations is 88 ° (in some cases 80°). When 0/4 = 90 °, Eq. (4) reduces to:

(E-' - 1)(sin201 - sin202)

- h E - l ( 1 - F ) ( s i n 01 - sin 02) = 0 ,

and for any value of h, 01 = 02 is a solution leading to the static solution W = 0 from Eq. (2). A finite value for h is insufficient to drive the wall across a defect of any width, and the coercive force, he, is infinite for 0 H = 90 °. This, of course, is equivalent to the 1/cos(0/4) result wherein, when the field is perpendicular to the easy anisotropy axis, the torque on the magnetic moment is at a minimum. The curves in Figs. 2 - 7 can be extrapolated to the value O H = 90 ° accordingly.

4. Results

Fig. 2 shows the ratio of the normalized coercive forces h c ( O / 4 ) and h~(0) versus the angle of devia-

/ f f v 5 /' , - i / (3)1 o

/ #27,- ZZ / /°.o(1)/ / o - - /

\ / /o,°.o" Z 1.5. o a

0.5 • 3JO 6~0 9 0

~H(degrees) Fig. 2. Ratio of the normalized coercive forces hc(On)/hc(O) as a function of the angle of the external field with respect to the anisotropy axes, OH, for a constant normalized defect width W = 2. Curve (1) is for E = F = 0.4, curve (2) is for E = F = 0.5, and curve (3) is for E = F = 0.6. Curve (4) shows the 1/cos(0 n) angular dependence generally accepted to be characteristic of a domain wall pinning-dominated coercivity. The solid circles are measurements on the angular dependence of FevvNdlsB8 [8].

tion, O H, of the external field with respect to the anisotropy axes. The normalized defect width W is held constant at a value of 2. The chosen values of the ratios E and F range from 0.4 to 0.6. These small values indicate large changes within the mag-

netic parameters A i, Ki, and Mi across the boundaries x = " D / 2 and x = D / 2 . Also included for comparison is the normalized angular dependence,

h c ( O n ) / h c ( O ) = 1 / c o s ( 0 n ) , generally accepted as the angular dependence curve characteristic of a domain wall pinning-dominated coercivity. We notice that our curves deviate significantly from the inverse cosine curve for this range of E and F and value of W with the deviation decreasing when E and F approach unity. Note the slight decrease in hc(O/4)/hc(O) as 0 n increases from zero. This 'd ip ' in the normalized angular dependence of h c is more pronounced for smaller values of E and F. Also included in Fig. 2 are measurements of the angular dependence of the coercive force for FevvNdlsB8 made by Kronmiiller et al. [8].

Fig. 3 differs from Fig. 2 in that E is now held constant at 0.8, while F = 0.4 and 0.6, corresponding to a large change in the magnetic parameters A i and or M i across the boundaries x = - D / 2 and x = D / 2 . The inverse cosine curve, hc(On)/hc(O) =

1 / c o s O n , is also included. We see that the increase in the normalized angular dependence of h~ as 0 n

B. Edwards, D.I. Paul~Journal of Magnetism and Magnetic Materials 147 (1995) 331-340 335

o

0

~m(degrees) Fig. 3. Same as Fig. 2, with E = 0.8 and W = 2. Curve ( l ) is for F = 0.4 and curve (2) is for F=0.6. Curve (3) shows the 1/cos( O H) angular dependence.

1 00

S \ / _ZZ

075- X X ° / o o\o / ° / / ' ~ " \ \ ~ o / o (2 ) / o , ----" a ~ "---0~0~.~-0 /o a

cb \ o ~ o '° <5)/ "% --o_ o-~ . . . . / .~-o

a - - - - - ~ a ~ a / a 0.50 I

30 610 90

~H(degrees) Fig. 5. Same as Fig. 2, with the normalized defect width W = 2. Curve (1) is for E = F = 0.1, curve (2) is for E = F = 0.05, and curve (3) is for E = F = 0.025.

increases is more marked as F approaches unity, with hc(OH)/hc(O) approaching the inverse cosine curve.

Fig. 4 differs from Fig. 2 in that F is now held constant at 0.8 while E ranges from 0.4 to 0.6,

corresponding to a large change in the magnetic

parameters A i and or K i across the boundaries x = - D / 2 and x = D / 2 . Again, we have included for comparison the 1 / cos (0 H ) angular dependence.

Here, the increase in hc(OH)/hc(O) with O H is more pronounced as E approaches unity, with this normalized angular dependence approaching the inverse cosine curve.

In Fig. 5, the ratios E and F take on very small values ranging from 0.025 to 0.1, and the normalized defect width W is again held constant at the value of

2. This range for E and F corresponds to differences

within the material parameters A i, Ki, and Mg between the host matrix and the defect region of up to

two orders of magnitude. The 'd ip ' in the

hc(On)/hc(O) plot referred to in our discussion of Fig. 2 is now very pronounced. Physically, the exter-

nal field required to drive the domain wall across the defect is decreased when applied in directions deviat-

ing from the anisotropy axes with the minimum

required field occurring at 0 n = 50 °. Note that for E = F = 0.025, applying the external magnetic field at 55 ° with respect to the anisotropy axes will reduce the coercive force by approximately one-half that required to demagnetize the material with a field parallel to the anisotropy axes.

Fig. 6 shows hc(OH)/hc(O) versus 0 u for small values of the normalized defect width W equal to 0.1

and 0.02. The ratios E and F are held constant at the

45 "y/ 0 5.5 / o °/°°(°f~ (J _c:

2.5. --~ / oS~,~o "°°1 -- 0/~

C-

0.5 310 6~0 9 0

8H(degrees) Fig. 4. Same as Fig. 2, with F = 0.8 and W = 2. Curve (1) is for E = 0.4, curve (2) is for E = 0.5, and curve (3) is for E = 0.6. Curve (4) shows the 1/cos(O H ) angular dependence.

C) ©

. .£z 4

r ~ o/

"~-J 2 ~ L . . . - o j ~

%]] o~O --I°/

0 J 0 30 60 90

8x(degrees) Fig. 6. Same as Fig. 2, with E = F = 0.03. Curve (1) is for W= 0.1 and curve (2) is for W = 0.02. Curve (3) shows the 1/cos(O H ) angular dependence.

336 B. Edwards, D.L Paul~Journal of Magnetism and Magnetic Materials 147 (1995) 331-340

6 J

o ©

i 7 -

"---1 2 (3

C - e, ; ,

° J0 6'o go

8H(degrees) Fig. 7. Same as Fig. 2, with E = F = 0.75. Curve (1) is for W = 2

and curve (2) is for W = 0.4. Curve (3) shows the I / c o s ( 0 N)

angular dependence.

small value of 0.03 and the inverse cosine curve is again given for comparison. Notice that as W approaches zero; the 'dip' (present at W = 2 in Fig. 5) goes away, and the curve tends to 1/cos(0H).

Fig. 7 also shows convergence to the 1/cos(0 H) angular dependence for decreasing W (W = 2, 0.4), but for larger values of E and F equal to 0.75. Here the convergence is quicker where the values of A i,

K i, and Mi for the host material and the defect differ by only a small amount. By inspection of Figs. 6 and 7, the proximity to 1/cos(0 H) achieved by W = 0.4 for E and F of order unity in Fig. 7 is not reached until W = 0.02 in Fig. 6 where E, F << 1.

Fig. 8 plots h c versus W rather than hc(OH)/hc(O)

versus 0 H. We do this for the values of O H ranging from 5 ° to 80 °. The ratios E and F are held constant at 0.5 and 0.75, respectively. Notice that for a given

0 . 8

o)2___.~ o

o/ (2) <") / /o / ,,~,,~'~ '~--"--":' ( - ~ ) '~

<-- 0.4.

0 . 0 , I , I 1 2

W Fig. 8. Normalized coercive force h c as a function of the normal-

ized defect width W for the values E = 0.5 and F = 0.75. Curve

(1) is for O n = 5 ° , curve (2) is for O n = 35 °, curve (3) is for

0 H = 65 °, and curve (4) is for O n = 80 °.

value of O N, when W is greater than approximately 1.5, there are essentially no further changes in h c.

5. Discussion

To better understand the physical mechanisms accompanying the different angular dependence curves found above, it is helpful to look at the effects caused by independent variations in the anisotropy, exchange, and magnetostatic energy constants K i, M i, and A i. In this way, one can better understand the degree to which each of the above energies contributes to the behavior of these curves.

Consider three different thought experiments in which we place defects having different magnetic parameters into a host medium. In all three of the 'experiments', we are considering the same host material having parameter values A 1 , K I , and M1; and the defect will always be the same width D. In the first 'experiment', we consider three magnetic materials containing different defects, the first having A 2 = 0.67A 1, the second having A 2 = 0.56A1, the third having A 2 = 0.45A 1 with all defects having K 2 = 0.89K 1 and M 2 = 0.89M v The normalized angular dependence curves derived from this 'experiment' can then be associated with Fig. 2 since E = F = 0.6, 0.5, and 0.4 for the three materials, respectively. The second thought experiment, with two different magnetic materials could be associated with Fig. 3. Here, the defect parameters would be M 2 = 0.67M 1 and M 2 = 0.45M 1 for materials one and two respectively with A z = 0.89A~ and K 2 =

0.89K~ for both. Finally, the third thought experiment would have three materials with defect parameter values K 2 = 0.67K1, 0.56K~, and 0.45K~, respectively, and A 2 = 0 . 8 9 A ~ and M z = 0 . 8 9 M 1 throughout. This 'experiment' would give a normalized angular dependence curve hc(OH)/hc(O) corresponding to that of Fig. 4.

Notice that changes in the defect parameters A 2

or K 2 will change the normalized defect width W as well as E and F (W= ( A 2 / K 2) 1 / 2 D ) . Therefore when we say that the thought experiments in the paragraph above lead to angular dependence curves corresponding to those in Figs. 2 -4 we are neglect- ing the changes in W caused by the variations in A 2

B. Edwards, D.I. Paul~Journal of Magnetism and Magnetic Materials 147 (1995) 331-340 337

and K 2 since W is held constant in Figures 2-4. However, including these variations in W produces no significant change in the analysis of the thought experiments. Indeed the minimum W these variations will produce is approximately 1.6 in the first 'experiment' when A 2 changes from 0.67A 1 to 0.45A 1 and all other deviations from W = 2 are to values greater than 1.6. As was seen in Fig. 8, the coercive force for all values of On undergoes no significant change for values of W in the range (1.5, oo). Since the changes in W 'unaccounted' for by Figs. 2 -4 occur only in this range, these curves do in fact correspond well to their respective thought experiments despite the fact that W is held constant at 2.

The magnets in each of the three 'experiments' mentioned above will have angular dependence curves which deviate significantly from the inverse cosine curve (as in Figs. 2, 3, and 4). This denotes the sizable effect that a sharp change in A i, g i , o r

M i across the boundaries x = - D / 2 and x = D / 2 can have on the angular dependence of the coercive field, with a greater change in Ai, Ki, or M i causing a greater deviation from the 1/cos(0 n) angular dependence.

Notice the symmetry with which the three 'experiments' above are set up. In all three of the situations, two of the defect parameters are held constant at (0.8) 1/2 times the respective host parameter and the third is taken a s 0.6/(0.8) 1/2 , 0.5/(0.8) 1/2 , and then 0.4/(0.8) 1/2 times its respective host parameter for the three magnets, respectively (Fig. 3 skips the 0.5/(0.8) 1/2 multiple). If all of the energy parameters played the same role in the behavior of the angular dependence, one would expect the three 'experiments' to produce the same curves. However, they are not the same. In fact, the inhomogeneities in M~ in Fig. 3 at F = 0.4 and 0 n = 88 ° produce a deviation from the inverse cosine curve of 83% as opposed to a deviation of 93% (Fig. 2, F = E = 0 . 4 , O n=88°), and 90% (Fig. 4, E = 0.4, O n = 88 °) due to inhomogeneities in A i and K~, respectively. The three parameters associated with these energies have different effects on the angular dependence of the coercive force, with the exchange and anisotropy energies playing a slightly more dominant role than the magnetostatic energy in the deviation from a 1 /cos(0 n) angular dependence.

The results for the three thought experiments follow what we would expect, given the arguments made for the occurrence of a 1/cos(O n) curve for values of H c << H A [4,6,8]. If the magnetic parameters have values such that the coercive force at O n = 0 ° is much less than the anisotropy field ( H A = 2 K J M I ) , in the host material (i.e. Hc(O°)/HA << 1), then an inverse cosine angular dependence will occur. In the analysis of the normalized coercive force for O n = 0 ° done by Friedburg and Paul [9], it was shown that Hc(O°)/HA goes to zero as E and F converge to 1, i.e. as the change in A i, Mi, and K~ between the host material and defect becomes negli- gible. Thus, the basic criteria for the 1 /cos(0 n ) angular dependence to take over is reflected in our three thought experiments. Also, using the data in Fig. 4 of Ref. [10] (an expansion on the findings in Ref. [9] by D.I. Paul), where 2Hc(O°)/HA is plotted as a function of E for different values of F, it is seen that indeed H~(O°)/HA increases as either E or F increases, with an increase in E and F simultaneously causing a rapid increase in H~(O°)/HA fol- lowed by a slightly smaller rate of increase caused by an increase in E alone, and finally the smallest rate of increase in Hc(O°)/HA is caused by an increase in F alone. Therefore, as in the three 'experiments', a change in A i between the host and defect material, causing an increase in both E and F, produces the greatest deviation from the inverse cosine curve while a change in K~, causing an increase in E alone, produces a slightly smaller deviation and finally the smallest deviation is pro- duced by a change in Mg between the host and defect material, causing an increase in F alone.

As seen in Fig. 8, at any given O n, h~(O n) as a function of W becomes essentially constant for W > 1.5. This asymptotic behavior in h~(O n) as a function of W was shown for 0 n = 0 ° by Paul [9]. We see now that it is exhibited at values of O n other than 0 ° and from this conclude that the behavior of an angular dependence curve, hc(On)/hc(O), is only a function of W for W less than approximately 1.5. Above those values of W, the host material regions on either side of the defect are sufficiently separated so that there are no further changes in hc(On)/h¢(O) if W is increased.

We see in Fig. 2 that there is a slight decrease in the normalized coercive force as O n increases. Fig. 5

338 B. Edwards, D.L Paul/Journal of Magnetism and Magnetic Materials 147 (1995) 331-340

shows this 'dip' more clearly for E, F < 0.1. Indeed, not only is the increase with O H shown in our model not as great as that predicted by the inverse cosine curve, but for E and F small ( < 0.5), hc actually decreases as 0 n increases from 0 °. Physically, this means that for these values of the parameters E, F, and W, it is easier in general to demagnetize the material by applying the external field in directions other than along the anisotropy axes. The maximum decrease in h c for very small E and F occurs at 0 n ~ 50 °.

In comparison with Fig. 2, where E and F simultaneously range from 0.4 to 0.6, little or no dip is seen in Figs. 3 and 4 where either E or F has the value of 0.4 - the other ratio being close to unity. It appears that both E and F must be small together in order for a significant 'dip' to be present. This suggests that the exchange energy takes on the dominant role in the cause of this 'dip'. Even in cases where either M 2 / M 1 <0.5 or K z / K 1 <0.5, the presence of such a dip still requires the ratio A z / A 1 to be small. Only when both K 2 / K 1 and M 2 / M 1 are simultaneously small is there no requirement that the ratio A z / A 1 be small.

It has been shown by Friedberg and Paul [9] that reducing the width of the defect region weakens the pinning strength of the defect and thus reduces the coercive force with respect to the anisotropy field in the host material. Therefore we would expect a smaller value of W to produce an angular dependence closer to that of the inverse cosine curve. Indeed, if W is small the two host material regions separated by the defect are then closer together and thus have more influence on each other eliminating the disturbance caused by the defect-region. How- ever, for a given width the amount with which the two regions can interact with each other is dependent on the magnitude of the changes in the magnetic parameters across the host -defect boundary (Az/A1, Kz/K1, M2/M1). One factor in particular that has proved to have a strong influence on the appearance of the 1/cos(0 n ) angular dependence is A z / A 1. We see in Figs. 6 and 7 that the changes in A i across the boundaries x = - D / 2 and x = D / 2 do have a large effect in determining that range of 'critical widths' of the defect for which Hc is sufficiently small with respect to H A such that we obtain an angular dependence curve close to I/CoS(OH).

Assuming that the changes in E and F are caused primarily by changes in A2/A 1 then, in particular, if this ratio decreases from 0.75 to 0.03, as seen in Figs. 6 and 7 (a factor of 25), the 'critical width', as shown in curves 2 of these figures is reduced by an order of magnitude. For widths larger that this 'critical width', the angular dependence has significantly deviated from the inverse cosine curve. Calculating Hc(O°)/HA at the parameter values and respective 'critical widths' (i.e. curves 2 in Figs. 6 and 7), and using Ref. [10] for values of He(0°), we get in both cases that Hc(O°)/HA ~ 0.02. Therefore, it appears necessary for H c to be approximately two orders of magnitude smaller than H A for the 1/cos(0 n ) to be a good approximation for the angular dependence, in contrast to the experimental values of 1 /3 to 1 /10 of H A [1,2].

The coercivity measured as a function of the applied field angle for Fe77Nd15B8 [8] has been included in Fig. 2 for comparison with our theoretical curves. Notice that curve one of this figure is in the best agreement with the experimental work although the fit leaves something to be desired, as was the case when this data was compared to other theoretical work [7,8]. There needs to be more experimental work done in this area including further knowledge of such parameters as A, K, and W in the defect region.

We now set out to explain the physical cause of the dip that occurs in our model for small values of E and F. In the formulation of the Stoner-Wohlfarth angular dependence model, the material is postulated to be a single domain such that the dipoles on the boundary of the domain are not fixed at any one given position. Furthermore, the movement of the dipoles in the material is coherent and the whole material can be expressed as one dipole M. There- fore, to reverse the magnetization of the material is to reverse the direction of M. Consequently, these postulates neglect the effect of any nucleation or propagation of domain walls within the material, thus making the calculations easier to carry out. When applying an external field h opposite M and at a small angle with respect to the anisotropy axes, the dipole M will move to a position of local minimum energy. When h = 0 that local energy well is parallel to the easy axes opposite h and thus M does not move. As h increases, this energy well

B. Edwards, D.L Paul~Journal of Magnetism and Magnetic Materials 147 (1995) 331-340 339

moves away from the easy axes towards h, disap- pearing finally when h reaches the critical field h c. At that point, M will flip to the other position of minimum energy; in the direction of h. As the angle of the external field increases, the same process occurs, although that energy well which is away from h disappears at a lower value of h. This lower value of h required to reverse the magnetization M is the dip we see in the Stoner-Wohlfarth curve.

Kronmiiller et al. [8] derived an angular dependence curve for a nucleation-dominated coercive force that used postulates similar to that of Stoner and Wohlfarth with the inclusion of the second anisotropy constant. Not surprisingly, this model also gave a curve of the normalized angular dependence that decreased as O n increased from 0 °.

Sakuma et al. [7] derived an angular dependence curve for a nucleation-dominated coercivity in a material containing a defect. Their model, based on that of Paul [10], took into account the spatial dependence of the magnetization orientation within the domain walls. This method, also used in the present paper, involved finding solutions to nonlinear differential equations. Because they were dealing with the case of domain wall nucleation, their postulates did not specify a fixed position for the dipoles at the boundary. Their curves show a dip for small E, as one would expect, since this approximates the Stoner-Wohlfarth model. However, holding W = 3 and F constant at either 0.5 or 1, their dip decreases as E goes from 0.2 to 0.1, unlike our result for domain wall pinning. No results are given for E > 0.2.

Our model also incorporates the spatial dependence of the magnetization orientation within the domain walls, but assumes a domain wall pinning- dominated coercivity. Therefore, our postulates in- clude fixed dipoles at the boundary. This leads to the occurence of the dip in the normalized angular dependence for small values of the parameters E and F. For these parameter values, the defect is effec- tively uncoupled from the host medium. Therefore magnetic reversal in the host medium would occur for values of the external field no greater than that required to nucleate a new wall on the other side of the defect. The decoupling of the defect from this other side of the host medium insures the free movement of the end dipole of that host region. Therefore

nucleation can occur similar to that of the Stoner- Wohlfarth model since there are no further defects in the material and one end is free to move.

The nucleation-dominated angular dependence curves derived in Ref. [7] as well as the pinning- dominated curves in this paper show the decrease in h c as a function of 0 n for certain parameter values. Although for given values of E, F, and W the model in Ref. [7] and our model may give entirely different looking curves; as a whole one can not look at an angular dependence curve to determine the coercive force mechanism without explicit knowledge of the parameters E, F, and W. These qualitative similari- ties are seen to be true when comparing our curves to nucleation-dominated angular dependence curves derived from other models as well [6,8]. This indi- cates that caution must be used if one is trying to determine whether a certain material's coercivity is pinning- or nucleation-dominated by comparing its angular dependence curve with theory.

Appendix

The algorithm used to determine the solutions to Eq. (4) was a combination of two searching proce- dures.

The first searching procedure, searchl, was per- formed by rewriting Eq. (4) with a quadratic polyno- mial, p(x), on the left and a semicircular function, g(x), on the right. To find a solution, we must then find an intersection of p and g. This is done by finding all of the zeros of p and g and thus subdi- viding the domain into regions where p or g is strictly positive or strictly negative. All of the intervals such that the sign of p is opposite to g are then discounted since there can not be a solution in that interval. The remaining portions of p and g are then translated towards the x-axes (by the same distance so as to preserve the points of intersection), and again subdivided into positive and negative regions so that the intervals over which these new curves have opposite sign can be thrown out. This process is repeated until there are only very small intervals left to search, (having length less than +0.004). The interval containing the greatest values is then chosen and the midpoint of this interval, Xm, is taken as a possible solution.

340 B. Edwards, D.L Paul~Journal of Magnetism and Magnetic Materials 147 (1995) 331-340

This first interval out of which we propose a solution may indeed not contain a solution. This first procedure is used since it never overlooks a true solution due to the fact that p and g are continuous. This next procedure, the second procedure, used because it never gives a false solution, could not have been used alone as it may overlook a true solution.

To insure the validity of the proposed solution from the first procedure, we take the original left side of Eq. (4) and evaluate it at eleven equally spaced points on the interval centered at x m having radius 0.004. The interval is known to contain a solution when the evaluated points take on both positive and negative values. Indeed, any continuous function taking on both positive and negative values over an interval contains a zero in that same interval.

If the second procedure does not insure a solution in the interval, we continue with the first procedure where we left off until we either throw out the interval in question or the second procedure insures a solution with the second procedure now evaluating at ten times the original number of points.

The resultant solution has accuracy of at least + 0.004. This was sufficient except for Figs. 6 and 7

where W << 1. In those cases we were working with very small numbers as values of the resultant integral (Eq. (2)), and it was necessary to refine the error to 0.00004.

References

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[3] D. Elbaz, D. Givord, S. Hirosawa, F.P. Missell, M.F. Rossig- nol and V. Villas-Boas, J. Appl. Phys. 69 (1991) 5492.

[4] D. Givord, P. Tenaud and T. Viadieu, J. Mag. Magn. Mater. 72 (1988) 247.

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ferromagnetic coercivity and applied field orientation

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