Journal of Magnetism and Magnetic Materials 73 (1988) 311-317 311 North-Holland, Amsterdam

DOMAIN STRUCTURE OF FERRIMAGNETIC GARNETS IN INHOMOGENEOUS PULSE FIELDS

K. PATEK Institute of Physics, Czech. Acad. Sci., Na Slovance 2, CS-180 40 Prague 8, Czechoslovakia

V.L. SOBOLEV and V.L. DORMAN

Physico-Technical Institute, dcad. Sci. of the Ukrainian SSR, Donetsk, USSR

Received 23 February 1988

The pulse remagnetization of art originally magnetically saturated ferrimagnetic garnet is investigated experimentally. The sample is a thin uniaxial layer with the easy axis of magnetization normal to the sample plane and magnetic pulses are produced by a straight wire positioned on its surface. The Slonczewski equations are applied to the description of the quasistatic behaviour of the nucleated domain structure. The experimental results are compared with theory and good qualitative agreement is obtained. For a more precise description of the dynamical behaviour the Slonczewski equations are extended to a case including spatial dispersion of relaxation due to the exchange interaction.

1. Introduction

Thin epitaxial films of ferrimagnetic garnets with the easy axis perpendicular to the plane represent very good model systems for the investi- gation of the behaviour of domain and domain walls. A large number of experimental and theo- retical works are devoted to the domain wall dy- namics, but so far there remains a great number of questions to be solved, as is the study of domain behaviour in inhomogeneous external magnetic fields. For the solution, new experiments and new theoretical methods are necessary and this is the scope of our interest. Such experiments include investigation of the remagnetization by an inho- mogeneous pulse field in the case of the originally saturated sample. If the in_homogeneous pulse field is generated by a wire located in touch with the specimen, the remagnetization occurs by spread- ing of the stripe domain structure in the direction perpendicular to the wire. This requires a theoreti- cal solution of the problem of dynamical and quasistatic behaviour of the domain structure in an inhomogeneous magnetic field.

In the theoretical part of our work an extension of the Slonczewski equations is presented and the equations obtained are applied to describe the pulse remagnetization mentioned above. In the experimental part the results of pulse remagnetiza- tion are presented. The interest is concentrated on the resulting equilibrium domain structure stable during pulse application and the theoretical and experimental results concerning this structure are compared. Comparison of the dynamics results will be reported elsewhere.

2. Theory

2.1. General dynamical description

In this part a short exposition of a generaliza- tion of the Slonczewski approach [1] to the do- main wall dynamics will be presented. The gen- eralization procedure consists of taking into account the spatial dispersion of relaxation due to the exchange interaction. The equations for do- main wall dynamics will be obtained in our work

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312 K. Pdtek et al. / Domain structure of ferrimagnetic garnets

on the basis of the Landau-Lifshitz equation with a relaxation term describing the spatial dispersion of relaxation [2]. The discussion of domain wall and magnetic soliton dynamics within this equa- tion may be found in ref. [3].

A more detailed description of this generaliza- tion and some results concerning the spatial dis- persion of relaxation will be given in ref. [4].

To derive the generalized Slonczewski equa- tions following the results on ref. [3] we shall consider the typical relaxation constant and longi- tudinal susceptibility of a ferromagnet as small parameters. This is justified because the dissipa- tive part of the Slonczewski equations [5] was obtained in the first approximation with respect to the small Gilbert relaxation constant. Thus we shall start from the equation:

th = - - y [ m , Heft] + YXrH ±

"[- "}t~ke [/n [/'/I , V2neff] ] , (1)

where m = M / M is the unit magnetization vector (M is the magnetization vector modulus), Heft is the effective field. The transverse and longitudinal components of this field with respect to m (calcu- lated in zero approximation with respect to the above mentioned small parameters) are defined by

H . = y - l [ m , th], HII = 8 W / S M = - 2 M - 1 W ,

(2)

where W is the total energy density of the system. Further, X r is the Gilbert relaxation constant de- scribing the relativistic relaxation mechanism, X e is the relaxation constant of the exchange origin, y is the gyromagnetic ratio and the dot on m denotes the derivative with respect to time.

Let us determine the coordinate system so that the x z plane is parallel to the domain wall and the z axis is directed along the magnetic easy axis; m will be characterized by the polar coordinates 0, ~p (fig. 1). Following the Slonczewski procedure [5] we shall consider

0 = 2 arctg exp( (y - q) A- l ) ,

~p = ~p (X, t ) ,

where k is the effective domain wall width and q is the position of the domain wall center. As a

Fig. 1. System of coordinates.

result from eqs. (1) and (2) we obtain the equa- tions:

8 o = _ 2M7_1{ 4 + A-lq [Xr + Xe/3 A2 + Xe~PZx] 8q

+ v H c sgn q} +X eJ , (3)

8o 8--7 = 2 My ' { 0 - A 4 [ X r + XjA2]

+aXe [ ~ - ½q5+2] }, (4) where o is the surface domain wall density, ~Px is the derivative with respect to the coordinate x and H c is the coercitivity field. The actual expression for A depends on the type of the crystal magnetic anisotropy which determines the internal structure of the wall ~p(x). The J term in eq. (3) arises due to the longitudinal component of the effective field and depends on the type of ferromagnet and on the internal structure of the domain wall:

J = 8A°°~°(~px x -- 2A-1Ax~px), (4a) 3A 2

where A is the derivative of effective domain wall

width and A 0 = ~ , o 0 = 4 A f t - 7 for a uniaxial ferromagnet (A is the exchange stiffness constant, K u is the uniaxial anisotropy constant). Eqs. (3) and (4) represent the generalization of the Slonczewski equations taking into account the spatial dispersion of relaxation due to the ex- change interaction. These equations correspond to the presently chosen domain wall model in which 0 = O ( y - q); ~p = ~p(x, t) and the domain wall center coordinate q does not depend on the coor- dinate x. A more general situation will be analysed

K. PZttek et al. / Domain structure of ferrimagnetic garnets 313

in ref. [4]. The account of relaxation dispersion is quite important if the dynamics of a domain wall with a complicated internal structure is considered as well as if the domain wall moving under the action of an inhomogeneous magnetic field is analysed•

We note, that eqs. (3), (4) are obtained in the same approximation as the Slonczewski equations [5]: Q > > I ; K u>>Kp; M H A < < K u;

I qq - l l >> IAA-1I,

where Q is the quality factor, Kp is an anisotropy constant of type other than uniaxial, and H A is a field perpendicular to the easy axis.

2.2. Quasistat ic specif ication o f the theory

In the experimental part the sample was origi- nally magnetically saturated and after that an inhomogeneous opposite magnetic pulse was ap- plied. We focus our attention on the spreading of the nucleated domain structure (fig. 2). In the following part of the work we will specify the theory for the particular case of the resulting quasistatic domain structure stable till the end of the pulse duration (see fig. 2d). The analysis of the dynamics will be published later.

As long as we are interested in quasistatics, we can exclude the time derivative ~ from eq. (3) and neglect its spatial derivatives. As a result we ob- tain

m *ti Ft°t = h--'--w - bd - 2 M H c , (5)

where a is the length of the stripe domains, Fro t is the total force acting on the domain wall located at the end of the spreading stripe domain, h is the thickness of the film, w is the width of the stripe domain,

m * = rn[1 + a2(1 + ~)]" m - ' d2° " ' ~ dq~ 2'

~.~ 2M yA x = - - ; b = ; ~ = a ( l + x ) '

3 A2~ r

Xr O L ~ - -

y M '

m* is the effective mass and/~ is the mobility•

/ j l z

J I H b

1-q b

Fig. 2. Spreading of the stripe domain structure nucleated by the straight wire located in touch with the specimen. (a) The sample is magnetically saturated by the external field /45; (b) nucleation of the reverse phase of the magnetization; (c) ex- pansion of the domain structure with reverse magnetization; (d) quasistatic domain pattern, stable till the end of the pulse

duration.

In the following calculation we will use a sim- ple model of domain structure shown in fig. 3. The total force in eq. (5) is determined by

dEto t F t ° t = - P d a '

where Eto t is the total energy per period p of the

314 K. Pdtek et al. / Domain structure of ferrirnagnetic garnets

I

o

I d a L. s ~. L Konec vzorku

Fig. 3. The model of the domain structure used in the theoreti- cal calculation.

stripe domain structure. The total energy includes three contributions:

E t o t = Emd + E w + E H , (6)

where Emd is the energy of magnetostat ic interac- tion, E w is the domain wall energy, and EH is the energy in the external field.

Explicit expressions for these energy terms are given in ref. [6]. Using the well known relations it is possible to express Fro t by means of an effective pressure/ ' tot and Ptot by means of effective mag- netic field //tot:

Ftot = etot hW, Ptot = 2MHtot.

Using the explicit forms of the energy terms [6] and the approximat ion h << a, h << L, a << L ( L is the distance between the ends of the stripe domain structure and the edge of the sample), we determine the contributions of the effective field corresponding to (6):

//to t = H(m°d ) - H 2 3 - n w - Hext, (7)

where

md -- ,tr3 h ](P( P , W ) ,

gm(1) 1 P ( 1 - - 2__ffW] h_, (8) a = 7M w p ] a

H w = ~rM 4l , W

W[l xp( O ( p , w ) = ~ T s i n ( ' ~ n p ) - P )1,

and l = Oo/4~rM 2 is the characteristic length of the material. Hex t consists of a constant bias field H b and the pulse field generated by a wire (figs. 2,3,4);

k I . 2 n e x t = H b rX/(X + 1 ) ; x = s / r , (8a)

s = a + d is the distance between stripe fronts and the wire (see fig. 3) in the equilibrium state, r is the radius of the wire, I is the ampli tude of the current pulse, and k = 2 × 10 3 0 e ~ m A -1.

Consequent ly eq. (5) takes the form

m *d = 2 M { H~0 ) + H~d ) -- H w - Hex t - H c ) - bd.

(9)

When the domain structure reaches the quasi- static state, the stripe ends are in equilibrium distance s f rom the wire and d = / / = 0 is valid. Thus to determine the value of s we solve the equation:

n(m°d ) - - n(mld ) - - n w - n e x t - n c = 0 . (10)

For the convenience of compar ison with the experiment we t ransform this equation to the form:

x B H b - H(m°~ ) + H w + Hc x 2 + 1 x - D Hp 0 , (11)

where

k I . D = d . B 2 H p 0 w r " H P ° = r ' r '

As a result of this calculation we obtained eq. (11) which allows us to obtain x and the equi- l ibrium distance s = xr, if p, w are taken f rom the experiment.

In fact the s value depends also on the distance d (see fig. 3). However, this dependence is very weak. The variat ion of 20% of d leads to a change of the s value of no more than 7% (in the case of d / s = 0.7). Thus it is possible to take the d value f rom the experiment or to calculate it on the basis of a simple concept ion (the boundary of a totally remagnetised area must be in such a distance d, where the module of the acting field Hex t is ap- proximately equal to the field H s ~ b for the stripe domain contract ion to the bubble form).

1(,. P6tek et al. / Domain structure of ferrimagnetic garnets 315

JOe] 12o-~

80- Hpx \

wire ~ ~ \ \ s~,mple surface ~-"~-'~-'~-'~-'~-'~J ~,....../_ 100 200 x [.,um]

_ . ,

Hp 80-

Fig. 4. The inhomogeneous pulse magnetic field produced by a straight wire. The curves describing the values of the perpendicular field component Hp and the in-plane field component Hpx a r e calculated for the pulse current amplitude 1 A.

In the theory we did not take into account the inf luence of the in -p lane componen t Hp~ of the pulse magnet ic field on the doma in wall. It is easily seen, that this inf luence is significant only in a short distance from the wire, i.e. just after the beg inn ing of the pulse at the init ial stage of do- ma in structure motion. The distance s is suffi- ciently large for H p J 4 ~ r M << 0.1.

Theoretical curves are shown in fig. 7.

3. Experiment

The experimental appara tus [7] uses the mag- neto-opt ical Fa raday effect and allows both stro- boscopic and high speed photography methods. The samples were uniaxial thin films of the ferri-

magnet ic garnets with the formula (Y,Sm,Ca)3 (Fe,Ge)5Oi2 with parameters described in table 1.

The experimental process was as follows. The sample was magnet ical ly saturated by an external

magnet ic field Hb, which was oriented along the easy magnet iza t ion axis (see fig. 2a). A n ad- di t ional inhomogeneous magnet ic pulse field Hp was applied opposite to H b by means of a pulse current flowing through the straight wire located in touch with the specimen surface (see figs. 2,4). The inf luence of the pulse field results in reverse

Table 1

saturation magnetization 4,nM = 211 G collapse field //co 1 = 104 Oe material length l = 0.53 I~m stripe period P0 = 9.3 ~tm film thickness h = 4.5 I~m coercivity field H c = 0.80e elliptical instability field H e = 75.10e contraction field Hs ~ b = 86.20e effective anisotropy field 2 K ~ / M - 4 ~rM = 610 G g-faktor g = 2.05

8 * ~i~ 2̧̧ ̧

Fig. 5. The photograph of the quasistatic domain structure which is stabilised in the time of the magnetic pulse applica- tion. The wire is located below (out of the figure area), the

reversed domains are visible as darker stripes.

316 K. P f t ek et al. / Domain structure of ferrimagnetic garnets

_ _ 5 10 15 20 25 30 Hb[ k-~-A]

5o3 40~ / ' \

i _ i i i

100 200 300 4.00 Nb[0e

Fig. 6. The dependence of the equ i l ib r ium dis tance s on the external field H b. The three curves cor respond to the three

values of the pulse cur rent I. Parameters of the curves: a) I = 1 8 A, b) 1 = 1 5 A , c ) I = 9 A .

magnetization nucleation and in the following spreading of the nucleated domain structure (fig. 2).

The created domain structure had the form of straight domain stripes perpendicular to the wire. During the pulse application the stripes are lengthening, this means the walls forming the stripe front are moving in the direction perpendicular to the wire. Because of the pulse field inhomogeneity (fig. 4) the velocity decreases with time and thus (for H b > Hcol) after a certain time shorter than the pulse duration the stripe domain fronts stop and an equilibrium domain structure in H b + Hp is formed. This domain structure is shown in fig. 5.

After the end of the pulse the structure col- lapses under the influence of the H b field and the sample becomes magnetically saturated till the next pulse.

The equilibrium distance s was showzl to be a reproducible parameter, characteristic for given values of the pulse current I and bias field H b. One set of experimental results is described in fig. 6 in the form of dependence of the equilibrium distance s on the H b value.

4. Discussion

The dominant influence causing the decrease of driving force and consequently of the wall veloc-

ity, is the inhomogeneity of the perpendicular component of the pulse magnetic field Hp. There- fore we will discuss the perpendicular component of the total field nex t acting on the wall in equi- librium distance s in which the stripe domain fronts stop (eq. (8a)). On the basis of measure- ment presented in fig. 6 we evaluate the depen- dence of the Hex t field on the H b value (for fixed I) . The resulting experimental curve 1 together with three theoretical curves 2,3,4 is shown in fig. 7.

The value of the perpendicular component of the total acting field nex t is shown to be a more sensitive parameter for testing and comparison with theory than the distance s. The theoretical dependence in fig. 7 are evaluated for three values of the stripe domain width w, because of the difficulty and uncertainty of the experimental de- termination of w from videorecording obtained by high speed photography. As the most satisfying appear the experimental values of p, w measured at equilibrium distance s, i.e. in the area of stripe fronts. In this case, in contradiction to the real situation, we neglect the increase of stripe domain width towards the wire; fortunately this is essen- tial only in the vicinity of the wire, i.e. at large distances from the area we are interested in.

If we take into account the uncertainty of the experimental determination of p, w and the

10 20 30 FIb[~-]

80

:£ 60 3

4O

1 2 2O

i

01 h i L i i 100 200 300 400 500

Hb[0e]

Fig. 7. The compar i son of the exper imenta l resul ts (fig. 6) wi th some theoret ical calculat ions. Curve 1 is the exper imenta l dependence, curves 2, 3, 4, are the theoret ical ones wi th p a r a m -

eters: 2: p = 20 ixm, w = 4 ixm, 3: p = 20 ilm, w = 2 p.m, 4: p = 20 ixm, w = 1 lam.

6

4

K. P&ek et al. / Domain structure of ferrimagnetic garnets 317

schematical form of the stripe domains used in the calculation (fig. 3), the comparison between theory and experiment represents good qualitative agree- ment and an explanation of the experimentally observed decrease of Hex t value with increasing H b value (fig. 6). For quantitative agreement it would be necessary to take into account the real form of the stripe domains. For a detailed theoret- ical calculation of the period p and domain width w it is necessary to solve the equations for the equilibrium domain structure in an inhomoge- neous magnetic field. It is clear that the problem can only be solved by numerical calculation. The theoretical dependence of the equilibrium distance s on the parameters H b and I based on such a more precise calculation in connection with ex- perimental results would allow us to determine the value of the characteristic material length l and so obtain a new method for the determination of this important parameter.

tion in equilibrium quasistatic conditions is com- pared with the experimental results.

In the experimental part of the work the related domain structure behaviour was studied. It was shown, that the domain structure nucleated by an inhomogeneous pulse field created by a wire is spreading in the form of stripes perpendicular to the wire. After a certain time the stripe fronts reach the equilibrium distance stable in the time of the pulse duration; this distance was measured and the magnetic field values at this distance are compared with the theory.

Qualitative agreement between theoretical and experimental results has confirmed the adequacy of the simple theoretical model of the domain structure which has been used. The way of im- provement was shown and the possibility of how to determine the material length parameter was proposed.

5. Summary References

In the theoretical part of the work an extension of the Slonczewski equations was derived for the case including spatial dispersion of relaxation due to the exchange interaction. This extension allows us in principle to analyse the domain wall dy- namics in nonuniform pulse fields. An explicit dynamic equation for the particular case of stripe domains generated by an inhomogeneous mag- netic pulse was derived. The solution of this equa-

[1] A.P. Malozemoff and J.C. Slonczewski, Magnetic Domain Walls in Bubble Materials (Academic Press, New York, 1979).

[2] V.G. Bar'yakhtar, Zhurnal exp.i teor. fiziki 87 (1984) 1501. [3] V.G. Bar'yakhtar, V.A. Ivanov, T.K. Soboleva and A.L.

Sukstanski, Zhurnal exp.i teor. fiziki 91 (1986) 1454. [4] V.L. Dorman and V.L. Sobolev, J. Phys. C, in press. [5] J.C. Slonczewski, Intern. J. Magn. 2 (1972) 85. [6] F.G. Bar'yakhtar, Yu. J. Gorobets, V.L. Dorman and V.L.

Sobolev, Phys. Metallov i metallovedenie 59 (1985) 1067. [7] I. Tom/t~ and K. Pfitek, Czech. J. Phys. B34 (1984) 1090.