Exact formula for the domain-width-dependence of the Bloch wall's magnetic moment in uniaxial ferromagnets

Volume 40, number 2

PHYSICS LETTERS

3 July 1972

EXACT FORMULA FOR THE DOMAIN-WIDTH-DEPENDENCE OF THE BLOCH WALL’S MAGNETIC MOMENT IN UNIAXIAL FERROMAGNETS R. STRAUBEL Central Institute for Cyberneticsand Information Processes, German A cademy ofSciences, Berlin, GDR

and W.J. ZI1~TEK Institute for Low Temperature and Structure Research, Polish Academi’ of Sciences, Wroctaw, Poland Received 23 May 1972

By applying the theory of periodic domain structures to a 180° Bloch ~all in a uniaxial ferromagnet, an exact formula for the dependence of the wall’s magnetic moment on the domain width is derived and numerically examined.

In many applications of the theory of ferromagnetic domains [1] particularly to magnetic filmd [2] the knowledge of the Bloch wall’s effective magnetic moment is quite important. The point is that, given the material and wall type, this moment M~(per unit volume of the wall) depends on the wall thickness ~ and the domain width the quantitative character of this dependence being neither theoretically nor experimentally known. In as mush as 5 itself is a function [3] of ~ (the latter depending in bulk specimens on the crystal thickness in the magnetically preferred direction [1, 4]), the wall’s moment M~is actually influenced by the mere domain width L~.In this note we show that, by applying the theory of periodic domain structures [3, 5,6] to the 180° Bloch wall in a uniaxial ferromagnet, one can easily derive an exact formula for the dependence of M~on ~ in spite of the involved relationship between S and z~. We utilize the periodic solution for the plate-like domain structure with 1800 Bloch walls, ~,

sinp=cnt,

t=2Kx/L~ K,

(1)

obtained in [5] from the variational principle for the angle p = p(x) which the magnetization vector in the yOzplane (Bloch wall plane) forms with the magnetically preferred axis Oz. Here, x is the variable along the direction normal to the wall, and K = K(k) is the complete elliptical integral of the first kind [7] which is a unique function of the modulus k, 0 ~ k ~ 1. Note that x n~,n = integer, corresponds in (1) to domain centers (at which p = nir) and x = (2n + l)~/2to wall centers (p (2n + l)z~/2).The formula for S and the relationship between k and ~ derived in [5] can be written in the form —

2kK=~/S0

(2)

where 5~= \/A7~isa material constant (A phenomenological exchange constant, K1 first anisotropy constant [1]). Let us introduce the reduced (dimensionless) magnetic moment where M0 is the saturation magnetiza tion per unit volume. Since only the y-component of the local magnetization vector contributes to M~we have =

~

f sin~

~

=

2SK

f cnt dt

=

arcsink

=

~arcsink

when utilizing eqs. (1) and (2). As k -*0 with ~ -~0and k asZ~_*0and,nw_*1 asz~—*oo.

-*

1 with

(3)

~

-*

°°,

it

is readily seen that rn~ 2/ir -*

0.637 IlS

Volume 40A. number 2

PHYSICS LLTTFRS

3 July 1972

2

I Ic 1

In a similar way one can obtain a lormula for the effective magnetic moment MD of the domain between two neighbouring 180 Bloch walls. In this case we have

mu

-

COS~(~

~

-

~

f

siitdt

In

+

k

ask 1. one easily when making use of (1). Noting [7j that K ~/2 as ~ -~ 0 and 2K 4 In 2 ln( I k2) proves that fl1 1) 2/~as — 0 and m~ -~ I as ~ [lie dependence of ni~and m13 on the reduced domain width ~/b0 is illustrated in fig. I. The curves are based on the representative numerical results m~= 0.64, 0.74. 0.92, 0.98, 1.00, 1.00 and m~— 0.64,0.69, 0.79, 0.86, 0.91. 0.96 obtained from eqs. (2)-(4) respectively lot ~ 1.00, 3.11, 6.67, 10.20, 15.90 and 29.71. Thus, with increasing domain width the moments ~ and m~attain practically (99%) their upper limit 1 (saturation) respectively for z~s 12 5~and .2s — 60 b0, while with decreasing ~ both quantities differ less than 1% from their lower limit 2/~for ~ be,. Hence the moments are sensitive to ~ only in these intervals. The corresponding total magnelic moments are then to be ~alcuIated from the formulae

1~ ~

~

.

‘~D— MOPODVE)

(5)

where fr~and V~are respectively the volume of the Bloch wall and the domain. Equally effective formulae for both moments can be derived for the various Bloch wall types in cubic ferromagnets. by utilizing the periodic solutions given tn [B]. Such calculations are in progress [9].

Ref erciices ]l] A. Seeger (editor), (hemische Bindung in Kristallen und I erromagnetismus (Springer-Verlag, Berlin Ne~York, 1966). [2] R.1 . Soohoo, Magnetic thin filnis (Harper and Row. New York, 1964): utton. Thin ferromagnetic films (Butterworth ~s1Prutton, liun ferromagnetic films (Butterworth and C’o., London, 1964): R. Streubel and Wi. Ziçtck, Phys. Status Solidi (1972), to be published. [3] J DIner, K. Durczewski and Vii. Liçtek. Acta Phys Polon. 35 (1969) 127 and 307 41 13. Wystocki and Wi. Ziçtek, P1~s. Lett. 29A (1969)114. [5] 3 Klamut and Wi. Ziçtek, Proc. Phys. Soc. 82)1963) 264. 161 W.J Zii~tek,Pliy~.Stat. Sol. 8(1965) 65. 7] P I . B3 rd and M.D 1 riedman, Handbook of elliptic integrals I or engineers and scientists (Springer-Verlag. Berlin-New York, 1971) [81 A. Waehniewski and Wi. Ziçtek, Acta Phys. Polon. 32 (1967) 21 and 93; 33 (1968) 581 [9] R. Odozynski, R Straubel and W.J Ziçtek. Phys. Stat. Sol., to be published. I I (~