On the magnetostatic energy associated with the variation of local magnetization direction in amorphous ribbons

~ ELSEVIER

Journal of Magnetism and Magnetic Materials 146 (1995) 137-142

Jeumalof magnetic materials

On the magnetostatic energy associated with the variation of local magnetization direction in amorphous ribbons I. Ciobotaru Institute of Technical Physics, D. Mangeron 47, lasi 6600, Romania Received 3 March 1994; in revised form 23 October 1994

Abstract

An expression is obtained for the magnetization dependence of the magnetostatic energy density associated with the volume magnetic charge density for a magnetized amorphous ribbon along the easy axis. The sample energy diminishes due to the occurrence of the domain structure. The ratio of the primary and secondary domain sizes, corresponding to the minimum energy, is calculated. Due to the made assumptions, the derived value of this ratio does not agree with the experimental data.

1. Introduction

Transition metal-metalloid magnetostrictive amorphous ribbons are characterized by small values of the constant of macroscopic anisotropy and relatively well defined easy axes. Due to the chemical and topological disorder, the tensor of local anisotropy shows spatial fluctuations. The orientation of the magnetic moments is determined by the balance between the local anisotropy interactions and exchange interactions [1]. Therefore an average direction and a certain distribution of the moment orientation results. The study of the domain structure by means of the Bitter technique renders evident, besides the large domains (parallel to the average direction), a system of secondary walls perpendicular to the average direction [1,2]. This system of walls is given by the distribution of the magnetic moment orientation within the domain. Due to the variation of the local magnetization direction, the divergence of the magnetization vector

is non-zero and therefore there will be magnetostatic interactions between the specimen neighboring regions. These interactions can play an important part in the magnetic and magnetoelastic behavior of the amorphous ribbons [3,4]. During the magnetization process the direction of the magnetic moments changes and the magnetostatic energy changes. Starting from the concept of pseudo-domains, Sablik and Jiles [5] obtained a magnetic coupling energy that changes in proportion with the second power of the anhysteretic magnetization. In the case of the amorphous ribbons this dependence was considered for small fields [3]. In this work the dependence of the volume magnetostatic energy density (associated with the volume charges) on the anhysteretic magnetization is obtained for a sample magnetized along the easy axis of magnetization. Since it is impossible to know the direction of the local magnetization, this is supposed to present a sinusoidal variation along the domain average magnetization. The volume magnetostatic

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138

L Ciobotaru /Journal of Magnetism and Magnetic Materials 146 (1995) 137-142 y

I

I

2Lti -?I I

I

! r I

O= y sin(~ry/l), f I I

r

I

vk

magnetization orientation, we shall make the following assumptions: (i) the vector of local magnetization M+ is considered of constant magnitude and changing direction; (ii) the magnetic moments are situated in the sample plane (xOy); (iii) the angle 0 between the local magnetization direction and the domain magnetization is a sinusoidal function of position, with the angular amplitude 3':

I J

2l

Fig. 1. Domain structure.

where l is the distance between two secondary walls; and (iv) the angle 0 is independent of the z coordinate. If L is the magnetic domain width, then the magnetization Ms(x,y) can be taken as a Fourier expansion:

Ms(x, Y) = Mo, o +

energy is reduced due to the appearance of the domain structure (as in the case of the magnetostatic energy produced by the surface charges).

2. The magnetostatic energy density associated with volume charges

(1)

Y'~

I

Mn,meX p i~r

+

m--/)l ,

n,m

(excluding n=m=0)

(2) where Mn, m are Fourier coefficients. The wave vector kn,ra is:

,3> The mathematical problem consists in finding the expression of the magnetostatic energy density associated with a certain volume charge distribution. We consider a 2 d thick amorphous ribbon with the other dimensions much larger than the thickness. We choose a rectangular system of coordinates with its origin in the ribbon center. The Oy axis is parallel to the average direction of the magnetic moments and Oz is perpendicular to the sample surface (Fig. 1). Given the chemical and topological disorder, the magnetic moment orientation shows a certain distribution around the average direction, with a certain angular amplitude "y in the ribbon plane [5]. A fraction of the total number of moments is oriented perpendicular to the sample surface. Thus, the mean domain magnetization, M D is smaller than the local magnetization M s. Since it is impossible to exactly know the local

We shall denote by Un,m the versor of the wave vector kn, m: Un, m =

kn.m/I kn.m[

Under these circumstances, the density of the volume magnetostatic energy is given by [6]:

Ev

tz° 2

( Un,m "Un,m)( Un,m "M-n,-m) fn,m, n~m

(excluding n=m=O)

(4) where /% is the free space permeability and fn,m is a factor considering the thickness effect:

fn,m =

1

1 - - { 1 2d k., m

-- exp( - 2 d kn,m) }.

(5)

139

L Ciobotaru/Journal of Magnetisra and Magnetic Materials 146 (1995) 137-142

We consider a domain pattern like in Fig. 1. Under the influence of a magnetic field parallel to the easy axis of magnetization (Oy), the sample volume fraction where the domain magnetization is parallel to the field will be v. Since 2L is the period along the o x axis, and 21 is the period along the Oy axis, and considering Eq. (1), the local magnetization components are obtained: • ,try M~ = Mssin[T sln(-~--)] , (6a)

My----~mscoS[3"s i n ( - ~ ) ] t ( x ) . The function

t(x)

1

t(x)

=

-1

(6b)

has expression:

where p = L/I and m r -- 2v - 1 is the reduced technical magnetization. For the demagnetized state, Eq. (13) becomes:

1 th(axp)}J~(3"). Ev(0, p ) = / ~ 0 M2 ( 1 - "rrp

The increase in the magnetostatic energy density during the technical magnetization when m r << 1 is:

Ev(mr,P)mr< 1 = E v ( 0 ,

p) + am~z,

(15)

where 27rp a = --/zoM:S22(3'). sh(2"up)

(16)

The increase in the magnetostatic energy density for a finite value of the magnetization is:

x~((2p-v)L, (2p+v)L), for x ~ ( ( 2 p + v)L, ( 2 p + 2 - v)L) for

(7)

AEv = e v ( m r, p ) - E v ( O ,

p)= a

(sh('rrpmr))2. ~rp

(17)

( p integer) From Eqs. (6) and (7) it follows that the non-zero components M~,m and MY,, of the Fourier coefficients are (Appendix Eqs. (A1) and (A2)): MoXo= (2v - 1)MsJo(3'),

(8)

M~," = -iMfl"(y), m odd integer, MY.,, = (2v - 1)MsJ"(3"), m even integer,

(9)

(10) 2

M y.,"= ~Mfl,.(y)

sin('rrnv),

m even integer, n4:0,

(11)

J"(3') are Bessel functions of the first kind with real argument• The technical magnetization is ( 2 v - 1)M D. By comparing this expression with Eq. (8) we obtain: M D = Jo(3')Ms
(12)

A simple expression for E v can be obtained taking into account that d -- l and supposing 3' ~ w / 1 8 (Appendix Eq. (A8)):

Ev(mr, P) { =/zoMs 2 1 × j2 (7),

(14)

Eq. (17) can be rewritten as: AE v = a

'

(18)

where a and /3 are material parameters.

3. Discussion The domain structure observed in certain amorphous ribbons in the case when the easy axis of magnetization is parallel to the ribbon axis can not be attributed to the surface charges or closure domains (for a long sample). In this case, an important part is played by the volume magnetostatic energy• The energy associated with volume charges will diminish by the advent of the domain structure. In the demagnetized state, the sample energy density is the sum of the volume magnetostatic energy density and the wall energy density:

Fo=lZo M2 _1 c h ( 2 7 r p ) - c h ( 2 ~ r p m r ) } "up sh~Z'-~p)

/3

(+ 1-

th(axp) j 2 ( y ) +

P-/

, (19)

(13)

where A is the exchange constant and K is the anisotropy constant.

140

I. Ciobotaru /Journal of Magnetism and Magnetic Materials 146 (1995) 137-142 ~0

E

J

I

,

,

,

,

,

v

secondary walls and not within the whole volume of the sample). Fig. 3 shows the increase in the magnetostatic energy density during the magnetization according to Eq. (18). During the magnetization along the easy axis situated along the ribbon, for long enough samples, the volume magnetostatic interactions will oppose the tendency to increase the volume fraction occupied by the moments parallel to the applied field H. The work of magnetization will be stored as magnetostatic energy. In this case, the energy density of the sample is:

I

~S

30

I

I

I

I

I

I

I

I

I

0.2

0,~

0,6

0,8

1

1,2

1,~

1,6

1.8

{ sh(flmr) ) 2 F(mr)

Fig. 2. Dependence of the sample energy density on the domain width.

Fig. 2 presents the dependence of the density of energy F 0 on the ratio p for the following values of the parameters: M s = 1.7 T; K = 102 j/m3; T = "rr/18; l - - 1 0 -s m and A = 5 × 1 0 -12 J / m . The energy F 0 presents a minimum for p = 0.6. From the experimental data it follows that p > 1 [1]. This disagreement results from the overestimate of the magnetostatic energy (perhaps the magnetic moments are changing their orientation only near the

2O

!

!

I

S

I

I

I

r

/3

- txoMDHmr .

(20) The reduced magnetization m r and susceptibility Xr will be: arcsh(2flh)

m,

,

2/3

(21)

Xr = (1 + 2/3h) -1/2,

(22)

where h is the reduced field I% MD H

h= - 2a

(23)

Figs. 4 and 5 present the field dependence of the

1

f

18

,r,

= Fo + a

i

i

i

!

r

,

i

0,8

16

0.6

14

O L,

12

0.2

10 8

-0.2 -0.~.

1

6

-0,6

~

2

-0,8 i

0 -0,8-0,6-0,~

-0,2

0

-I 0.2

0.4

0.6

0.8

-'

I

I

I

I

I

I

I

i

I

-~

-3

-2

-1

0

1

2

3

4.

mr

Fig. 3. Dependence of the density of the volume magnetostatic energy on the reduced magnetization.

Fig. 4. Dependence of the reduced magnetization on the reduced field given by Eq. (21).

I. Ciobotaru /Journal of Magnetism and Magnetic Materials 146 (1995) 137-142 1

141

Appendix

0.9

In order to calculate the Fourier coefficients, one must notice that Eq.(6) can be rewritten as:

0.8 0.7 0,6

m odd

0,S

My(X,y) = Y'. MsJm('y) exp(i~7~)t(x),

0.4

m even

0,3

(A1)

0,2

where

0,1 I

0 -'

-~

I -3

I

I

I

-2

-I

0

I 1

I 2

1 3

t 4

5

h Fig. 5. Dependence of the reduced susceptibility on the reduced field given by Eq. (22).

normalized magnetization and susceptibility given by Eqs. (21) and (22).

t(x)

t(x)

may be expanded as: 2 [. ~rnx ~ = ( 2 v + 1) + • - - s i n ( ' r r n v ) e x p [ t - - ~ ) . n.0 -rrn

(A2) For the energy calculation it is necessary to evaluate the expression (summation over n and m) E(Un,rn'mn,m)(Un,rn'm-n,-m)fn,m ~,m

= E {(UX,m)2[MX,ml2 +

(u.~,,.)2l M.Y,ml2

n,m

+2uX,,nuY,mRe(M:,mMY_._m)}f.,,., 4. Conclusions In the case of magnetization along the easy axis (parallel to the ribbon axis) of the ribbon with large length, the domain structure can not be explained by the appearance of the superficial charges or the closure domains. An important part is played by the volume magnetic poles due to the changes of direction of the local magnetization: - t h e domain width is determined by an interplay between the domain wall energy and the volume magnetostatic energy, and - during the magnetization process the volume magnetostatic energy increases. The expression for the magnetostatic energy density, Eq. (13), is only qualitative since it overestimates the magnetostatic interactions by considering changes in the direction of the local magnetization of the form (1) and for high values of the specimen magnetization the domain width depends on the magnetization. For small values of the magnetization, one obtains AE v at a m E. The domain magnetization M o has also been overestimated, since the magnetic moments directed perpendicular to the sample surface were neglected.

(A3)

where u.~,,., UYm, MX,m and M.Y. are the components of u.,,. and M., m. Taking into account the expression of the Fourier coefficients given by Eqs. (8)-(11) and of the versor U.,m, it follows that the first and the last terms on the right side of Eq. (A3) are vanishing. The coefficients M.,,. do not change when changing the signs of the indices m or n. The second term can be written in the form: E E ( UnY,m)2[ n~0 m~0 even

M:,m[2f~,m

= 2 E (u0,m)21M0'ml2/0,m rn~0 even

+4 E

y

2

E (U.,m) I mY,m]Zf.,m.

(A4)

m>2 n>_ I even

Thus, the magnetostatic energy density

Ev(m r,

Ev(mr, p)

is:

p) = /x0M2(2v - 1) E jm(~/)f0,m2 m>2 even

2 8

p2m2

+P'oMs-~ E

E

m>_2 .>_1 n2( even

×j2(y) sin2(~nv)f.,,..

+

(A5)

1. Ciobotaru/Journal of Magnetism and MagneticMaterials 146 (1995) 137-142

142

f,,,m

Since l = d = 10 I~m, it results that > 0.9. We fn,m = 1. Summing over n in the second term of (A5), by using the identity approximate

z2

Ev(mr,p) on

the reduced magnetization m r and the

ratio p is:

Ev(m~, P) 1 ch(2'rrp) - ch(2'n'pm~) /

n>l n2( n2 q-z2) sin2(~nv)

=/.zoMs 2 1 - ; p

sh~-~p)

,IT2

-

2

v ( 1 - v)

XJ~,(y).

(A8)

,IT2

4z s h ( c r z ) {ch(~rz) - c h [ ( 2 v - 1 ) r r z ] } , (A6) we obtain:

Ev(mr, P) = tZoM2 { 2 ch('rrmp)-ch('rrmpmr) ) E

m~2

1

even

XJ~(y).

1Trap

--~(-~

(A7)

Since j 2 ( y ) << j22(y) for m = 4, 6 . . . . . we consider only the first term (m = 2). The dependence of

References [1] A.P. Thomas and M.R.J. Gibbs, J. Magn. Magn. Mater. 103 (1992) 97. [2] P.T. Squire, A.P. Thomas, M.R.J. Gibbs and M. Kuzminski, J. Magn. Magn. Mater. 104-107 (1992) 109. [3] H. Chiriac and I. Ciobotaru, J. Magn. Magn. Mater. 124 (1993) 277. [4] H. Chiriac, I. Ciobotaru and S. Mohorianu, IEEE Trans. Magn. 30 (1994) 518. [5] M.J. Sablik and D.C. Jiles, IEEE Trans. Magn. 29 (1993) 2113. [6] A. Corciovei and Gh. Adam, J. Physique 32 (1971) C1-408.