2816 IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 5, SEPTEMBER 1990

DOMAIN WALL DYNAMICS AT THE Tc VICINITY L.V. Panina, D.A. Garanin, I.G. Ruzavin The Institute of High Temperatures, USSR

Academy of Science, 127276, Moscow, Izhorsky str.,13/19, USSR

Abstract - A new approach for classical description of the dynamics of magnetic systems in presence of thermal agitation has been proposed. The relation between longitudinal and transversal relaxation frequencies are obtained. At the vicinity of Curie temperature the domain wall structure and propagation have been investigated.

Introduction The temperature dependence of magnetic domain

propagation is of great practical interest, because it essentially determines the operation temperature range of magnetic memory devices. Current theoretical works on the domain dynamics are based on the micromagnetic theory. The basic assumption of this theory is that the magnetization magnitude is a function of temperature but not of space. It breaks down [1,21 near Curie temperature Tc, so the well-known expressions for domain wall velocity v t3.41 at the proximity of T are not applicable. In the present paper an approach taking into account thermal agitation is proposed. This method yields the non-linear differential equation for an average magnetization vector. Domain wall structure and expressions for v derived from it are discussed.

The micromagnetic theory involves the presence of thermal agitation, that becomes higher as the temperature approaches to the Curie temperature, using some auxiliary potentials as a function of the magnetic magnitude only [51 or expansion for the free energy of ferromagnet as a power series in magnetization (that corresponds to well-known Landau-Ginzburg approach [ 6 ] ) . However: these nnnthds do not yield all the necessary characteristics of magnetic dynamics near T .

Here, we consider the presence of thermal activation following the methods developed in the theory of stochastic processes [71. The thermal averaging leads to the Fokker-Plank equation for the distribution function of magnetic moments, which is similar to the equation for the density matrix in quantum mechanics. Using a special type of the distribution function, we show how it is possible to find the dynamics of average magnetization.

In comparison with the standard methods mentioned above our consideration allows to investigate magnetic dynamics when the magnetic magnitude is a function of space and temperature.

Landau - Lifshitz - Bloch tvDe eauation

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For the classical description of dynamics of a magnetic subsystem interacting with thermal background one can use the Landau-Lifshitz equation [31 with random parameters.

The influence of thermal activation can be described in terms of casual perturbations of magnetic field and magnetic anisotropy. Following this way we shell consider the model of +two anisotropies with constant anisotropy direction 6 and fluctuating one, described by a fluctuating anisotropy tensor K::(t),

and regard that every macroscopic volume of magnetic media will be affected by fluctuating Gfl(t) and external fiextmagnetic fields. Such a macroscopic magnetic volume is characterized by a magnetic vector G, with its magnitude Ms being independent of its orientation in space. Thus, the free energy of the svstem under consideration is:

The "Langevin equation" of the process is :

d; - [;;.(F; + F;f'(t))] - g; rMs dt aF

2 = - 1 - 0

ut a: ' G f l = ;;f'(t) + $"(t).;; .

Here a is a nonuniform exchange constant, D is a constant of uniaxial anisotropy, r is a gyroscopic parameter and 6 is a relaxation term. In the most typical cases thermal fluctuations have correlation times T~~ that are much shorter than response times of a magnetic subsystem rm. The value w m -l/rm is of the

order of gyromagnetic resonance frequencies wr< 10" and, according to the Nyquist formula, w f l - l/r f l = Wfi

(8=kBT is the thermal energy, fi is the Plank

constant). At room temperatures ~ ~ ~ - 1 0 ' ~ . so that the inequality 7 f l ((7 m is fulfilled in a wide temperature

range. AS z r.jx:t t1-2 fluctuational processes qr'(t) and K~~ !t) have a "white" spectrum and may be considered as delta-correlated processes:

<q:l (t)q;'(t\> = 2h i J -3mB((t-tjyMs)

equation (1) by the Fokker-Plank equation for the distribution function f(;,t) =< - ;(t)) >, Using standard methods, developed in [71:

A stationary solution of equation ( 2 ) must be a Hibbs type that determines the form of the relaxation vector R:

(3)

It is to be note that a relaxation term cannot be chosen arbitrarily as adopted in general fenomenological consideration and strongly defined by expression (3 . Only in the case of isotropic fluctuations ( f i is a diagonal tensor.) the relaxation 6 is reduced to the standard Landau-Lifshitz type. For the sake of simplicity we shall deal with this case. It means that Fokker-Plank equation ( 2 ) transforms to the "standard" form:

0018-9464/90/0900-2816$01 .OO 0 1990 IEEE

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'iJ = '16iJ* ' i k , J l = A 6 2 i J 6 k l ' ' i , k l = O' G = AaiJeS , h = A + A .

i J 1 2

This approach is analogous to the method developed in the theory of superparamagnetism [81. Equation (4 ) . in general, is complicated by the presence of gyroscopic terms. To solve (41, we employ a special type of the distribution function proposed and discussed in 191:

The introduced characteristic vector <(;,t') in parametrical presentation ( 5 ) plays a part of "mean field" (in stationary state it coincides with +Cff= <If(;)>; the average magnetization ;(g) = <p> =

S ; exp($.;)d;/Z is oriented along Such assumption leads+ +to the closed dynamic

non-linear equation for m(<). It is convenient to write this equation in terms of <:

direction).

Equation ( 6 ) describes longitudinal and transversal magnetic relaxation wJth the relaxation frequencies r and r depending on 5 . These frequencies tend to the same limit 2Aem at <a1 (T+Tc). This equation may be called as a Landau - Lifshitz - Bloch (LLB) type one. In the cases of low and high temperatures it coincides with Landau-Lifshitz (if the distribution function is rather concentrated near some direction) and Bloch equations respectively. As it was shown in [91, this method yields sufficient realistic approximation in a wide system parameter range (except a special case of large single-ion anisotropy presence).

Domain wall structure and propagation A typical dynamic task with the essential presence

of the longitudinal relaxation of magnetization is a problem connected with the propagation of domain walls at the T vicinity. The standard method, based on the Landau-Lifshitz equation, neglects longitudinal relaxation completely, so it becomes unreliable at the Tc neighborhood. Otherwise, using the method developed

above, it is possible to find domain wall structure and its dynamic properties near T correctly. Domain wall structure corresponds to the stationary solution of ( 6 ) with proper boundary conditions:

1 2

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2 = I f e f f ( < o ~ / ~ m , $Ay=fm)=f{ . ( 7 )

Fig. 1 Change of average magnetization vector ; along the

domain wall.

We shall consider 180' Bloch type domain walls. The average magnetization inside domains and easy anisotropy axis (D>O) are oriented along 2-direction as it is shown in Fig.1. Equation ( 7 ) has two types of solutions: a linear domain wall type (LDW) with c,=O, and a type named elliptic (EDW) with Q+O (ez and e, are the

orthographic projections of 2 on 2- and X-axes respectively). At the Tc vicinity and D/Bcal these solutions coincide with those, derived in [ l ] :

EDW with width aZ=a/D:

LDW with width a2=2a/(3ecT)

( 8 )

Here T=(T-T)/T :l, <:=15~, r*=2D/3eC, cos(p= th(y/a). Note, that parameter Q does not correspond to the "orientat'ion" angle P in Fig.1; the latter in EDW structure is connected with 9 by the expression t g d 1 - lOD/B 5' tgp.

c m For the sake of simplicity only these states and

their dynamics will be discussed below.. The EDU structure disappears at T=T while LDW one

exigts at ail temperatures but being inetastable at T>T . If T<T the only.existing type is linear with the width depending on r,*but coinciding with the width of the EDW type at T=T .

At relatively high T, when D/(ecT)al (i.e., when

the thermal energy is much less than the wall energy), the EDW type coincides with the well-known "circular" solution of the Landau-Lifshitz e?.'atix:

(10)

with parameter Q corresponding to the angle B in the Fig.1. These changes of domain wall structure with increasing of temperature is shown in Fig.2.

LDW LL

0 T

Fig. 2 Change of domain wall structure with increasing of

temperature. When appl ing external magnetic field some

disturbance a{ and Si?'' of the stationary state occurs. The simplest form of the dynamic solution of the LLB equation corresponds to domain .wall stationary motion with a constant velocity v. Assuming 6E/eoal, heXt /eoamal, where Eo is a stationary vector, we

obtain the equation:

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This system can be reduced to the form: -3 -3 SE = $(go,v,Cext), aE(+m)=O, (12)

where e is a matrix-differential operator. Because of the invariability (11) for shifting along Y axis, the function

g=ago/a: (13)

satisfies the equation: e.;; = 0 . (14)

bccording to the Fredholm alternative theorem [lo], to resolve edge problem (12). the function must satisfy the condition: (i*$)L=O. Thus, in linear field

approximation, if we know the stationary state, we can find the value of domain wall velocity v from the algebraic relationship:

2

*m

It gives :

v = r H e X t X / G -

(15)

, T<T

I 2aEDW , T L t (16)

102

10-1 10-3 100 103

Fig. 3 The temperature dependence of mobility ~ = v / ( i H ~ ~ ~ a ~ ~ ~ ) in comparison with Landau - Lifshitz one (dashed

curves) at various U ( U =0.5, U =0.21

At large values of the parameter p (p~l), when the "circular" structure exists, expression (16) coincides with the standard one, following from the Landau-Lifshitz equation solution:

v~~ErHeX 'ED w 2hA + 4h2), A2=1sT . (17)

At T=T function v(r1 (16) achieves its minimum. The difference between expressions (16) and (17) depends on the relaxation parameter A and may be considerable for a properly small h (see Fig.3). Then there is a sharp increase of V-lh instead of v-1/T1'2 following from (17). As it can be seen from Fig.3, obtained dependence (16) has a pscullar point with a first derivative breakdown at' T=T , when the symmetry of the system changes.

Note that in the case of, T<T (LDW type) domain

wall velocity can be estimated using more simpler but less general way. Because of the fact that in the linear wall the only longitudinal relaxation is possible, there exists a dynamic equation of a Bloch type: a<F> (mZ 1 _ _ dmz - - r-.

am* dt

Integrating this equation over the wall area and using expansion for <F>(m 1 in the series in power m (that corresDonds to well-known Ginzburg-Landau approach near Tc[6]) with stationary LDW type solution (9) of wall profile, it is easy to obtain (16).

Thus, our dynamic approach describing the general case of the two relaxation types agrees with the simple limits of thermal fluctuations absence and with presence of longitudinal relaxation only.

In conclusion, we should note that the discussed dynamic effects may be experimentally observed in some uniaxial ferromagnets with a relatively large value of the parameter T -D/B i.e., in ferromagnets with large anisotropy energy D and low Tc, for example, in rare-earth metal compounds.

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Conclusions The magnetic dynamics in the presence of thermal

fluctuations is approached here by methods familiar in the theory of stochastic processes. The closed non-linear differential equation for the average magnetization has been obtained. The relation between longitudinal and transversal relaxation frequencies has been derived. The domain wall pr'opagation has been investigated and new critical behavior of the wall motion has been explored.

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