06 - stability and probabilistic description of dw motion

6. S t a b i l i t y a n d P r o b a b i l i s t i c D e s c r i p t i o n

o f D W M o t i o n

Among various nonlinear effects arising in the DW dynamics an important place is occupied by the nonlinear dependence of the friction, Fd, on the wall velocity. The nonmonotonous Fd (v) dependence caused by Cherenkov phonon emission, which was discussed in the previous section, results, formally, in the ambiguous dependence of the DW velocity on the driving field. This raises the question: which one of the possible velocity values is realized? This chapter deals with an analysis of this problem.

We first study the DW stability relative to small perturbations and assure ourselves that, although some of the velocity values respond to the unstable motion, a complete answer to this question cannot be given. The analysis for the case of arbitrary (not small) perturbations makes it possible to solve, ultimately, this problem. It then turns out to be possible to explain the principal laws: the absence of hysteresis on the experimental behaviour of v ( H ) in the nonlinear region, and the weak temperature dependence of the magnetoelastic anomaly width.

Let us examine non-unidimensional DW motion, treating it as a mem- brane with surface energy a and mass m.. The equation for free bending vibrations of the wall can be obtained from the Lagrangian:

:/ { (0:),} s = -~ d y d z m , \ O r ] - o - ~ r 2 ' (6.1)

where m, is the effective mass of the wall, cr is its surface energy, m , c 2 = o-, and r• is the coordinate in the DW plane. It was assumed in (6.1) that v<< c.

The equation for the wall displacement f ( x , z , t ) , with allowance for the induced force FH and the retarding force, takes the form

or2 \oy. + Oz~/= .~--: -~ - .~. W (6.2)

Here, r] is the viscosity coefficient determining the usual viscous friction of the type F = -zlv, # is the DW mobility responding to this viscous friction, F ( v ) is the phonon retarding force. The (6.2)-type equation was investigated at the beginning of the century in connection with the analysis of string oscil- lations. We assume that the phonon frictional force obeys the same equation

88 6. Stability and Probabilistic Description of DW Motion

as tha t in the one-dimensional case. As it was noted in the previous section, such substitution, generally speaking, is invalid but for linear DW-type perturbations one may hope for its adequacy.

f = v0t, corresponds to a stationary motion of a straight DW, the velocity value v0 is found equating the r.h.s of (6.2) to zero:

rl(Vo - # H ) + F(vo) = 0 (6.3)

We examine small deviations from the stationary motion, writing, to this end, f = rot + qo, and linearizing (6.2) in ~:

Here the following notation is introduced: 2u(v0) = ~/+ dF(vo) /dvo . We Iook for a solution for 9 of the form u0 exp(Tt + i k • 1 7 7 For the increment 7 we get

-y = • V/ .2(vo) - d k l (6.4)

It follows from this formula that when z /> 0 the real part of 7 is negative, i.e. small deviations from the solution f = rot are damped. If, on the other hand, the velocity v0 is in the range of negative differential mobility, and ~(v0) < 0, the solution f = rot is unstable. The most intense increase is observed in inhomogeneous deviations of the DW velocity and shape from those corresponding to the straight and uniform motion, clk• [ ~ lu(v0)l corresponds to a maximum in the increment ~/ (Zvezdin et al. [6.1], see also the review by Bar 'yakh tar et aI. [6.2]).

Evidently, this regime can only be affected in the presence of a nonmonotonous part of the function Fd(V) = ~v + F(v ) , and with such mag- netic field values, when the equation (6.3) for v has three roots. It is readily seen that instability is typical for the mean root v~s, and two other roots vl < vus < v2 correspond to the stable motion with respect to small perturbations. As it will be shown below one of these solutions has absolute stability, and the other one is unstable with respect to not small perturbations.

We estimate the WFM parameters at which the inequality u(v0) < 0 is satisfied and instability develops. This inequality is satisfied at small enough r/, i.e. large mobility values, specifically, # > P0,

#0 = 2m0 {max IdF(vD/dvl} - t ,'. 2moz2V/Fmax (6.5.)

Here A v and Fmax are, respectively, the width and phonon peak height in the force of friction. Assuming that AHmax ----- Fm~x/2m0 ~ 30 Oe, and A v ~_ 0.2 s ~-- 105cm/s we get: P0 ~-- 103cm/s Oe. This value is consistent in the order of magnitude with the mobility value starting from which the nonequ- ilibrium and non-unidimensional DW dynamics is observed in orthoferrites, see below, Chap. 8.

6. Stability and Probabilistic Description of DW Motion 89

To s tudy the stability, with respect to not too small perturbations, is much more complicated, and so far, this analysis has only been done for the case of unidimensional DW motion. In this case f = f ( t ) in the equation (6.2). Following the Gomonov et al. [6.3] paper, we write equation (4.2) in the form:

0]9 + p = 2moll + F(; ) (6.6) Ot T

where p is the DW momentum, and in the actual case of motion with velocity v N s << c it can be assumed tha t p = m,v. The terminology used in Eq. (6.6) is: F(p) = F(v) = F ( p / m , ) .

To examine the stability of the s teady-s ta te motion with the given value P = Po, po/'r = 2mol l + f'(p) we may use the classical Lyapunov theorem on the stabili ty of motion, see, e.g., Ref. [6.4]. To make use of this theorem, in our case, it is necessary to construct some function ~(p), whereby near the point p = P0 this function

1) would be positive-definite at all p r P0; 2) would have a negative derivative with t ime calculated according to the

equation of motion (6.6).

We show, following Gomonov et aI. [6.3], tha t as the Lyapunov function we can choose the function

p2 if0 p ~(P) - 2~- 2 m o p H - P(~)d~ (6.7)

Since the equation of motion (6.6) can be writ ten as

ap 0 '

the derivative Or = - [0[~(p)] lop] 2 < 0, this satisfies the second conditon of the Lyapunov theorem. Our further analysis is reduced to studying the ex t rema of the Lyapunov function.

Indeed, the p ~ Po values corresponding to the s teady-s ta te DW motion are determined by the condition O~(p)/Opo = 0, i.e., correspond to s tat ionary points of the Lyapunov function. Considering the dependence of F on v for a single phonon peak at v = s, it is easy to see, that depending on whether # < Po or p > #0, the function ~5(p) has one or three extrema. Since the integral f o / ~ ( ~ ) d~ is finite, the function ~(p) ~_ p2/2T when p -+ c~. Thus, if there is one extremum, it is a minimum. If, on the other hand, qS(p) has three ex t rema at the points Pl < Pus < P2, then when p = Pus the maximum is affected, and when p = Pl and p = P2 - two minima.

If ~P(p) has only one ext remum for p = Po, then to the latter corresponds a stable motion with respect to arbi t rary perturbations. Indeed, the function ~(p) cc p2 when p --+ • and O~/Op vanishes nowhere except for the point P = Po. Hence it follows tha t at any initial condition for t -+ ~ , p --+ Po.


The motion with momentum Pus = m.v, as it has been shown above, is unstable (in virtue of the Chetaev theorem on the instability of motion, see [6.4]; it follows also directly from the fact that O~/Ot < 0 and 02~(pus)/Op~s > 0). As for the motion with velocities Vl = pl /m. and v2 = p2/m., in virtue of the conditions of Lyapunov's theorem, these are stable relative to sufficiently small perturbations. This stems from the fact that the functions ~(p) -4~(pl) and ~(p) -~(P2) are positive-definite in some finite vicinities about the points P = Pl and p = P2. The conclusion on the global stability of these motions, or one of them, cannot be made: one can always represent the perturbat ion that transfers the system from the minimum p = Pl to the minimum p = P2, or vice versa. The problem with what velocity the DW will move in the given field, under the condition that it initially had some velocity, is equivalent to that just to what minimum of the function ~5 the system will be transferred. From the view-point of mechanics these minima are equivalent. But from the view-point of thermodynamics there arises inequivalence, and the state of the motion to which the deeper minimum of the function ~5(p) corresponds, is preferable.

Varying the governing parameters, e.g., the external field, results in chang- ing the function ~5. The initial global minimum that determined the state of the system can then become the metastable local minimum or even vanish. In this case the system should go over from one local minimum to the other. To determine the moment of the transition and the minimum in which the state of the system will be stable (in the works on the DW dynamics this principle is called the principle of maximum retardation) was accepted implicitly. It can be formulated as follows: the system being, initially, in a given local or global minimum, remains in it until it is existent. This assumption gave rise, in the dynamical theory, to a conclusion inconsistent with experiment about the existence of hysteresis of velocity in the system.

This assumption does not take into account the availability of noise - fluctuations which are, certainly, present in the system, such as a DW moving in a real inhomogeneous sample. To take fluctuations into account, the random force _F(t) should be added to the r.h.s, of Eq. (6.6). In this case the equation (6.6) takes the form:

Op a~,(p) at Op

- - + . F ( t ) ( 6 . 8 )

where the mean value of/~(t) is zero, (F) = 0. Equation (6.8) has the sense of the Langevin equation. Unlike the purely dynamic equation (6.6) it describes the statistical rules, in particular, irreversibility, see, for example, Isihara's [6.5] monograph. In this case the character of the DW motion is determined by a function of the distribution of the momenta w(p, t). Under general as- sumptions upon the character of the statistical dynamics the kinetic equation of the Fokker-Planck-type can be obtained for this function (Gomonov et al. [6.3]);


aw O (wOq5 ~ 02 at -- Op \ -~p ] +-~p 2(Dw) ' (6.9)

where D is the diffusion coefficient. This magnitude characterizes the noise level in the system and determines the correlation function of the random force (~'(t)['(t')) = 2Db(t - t'). (Remember that the quantities 'p' and 'D' refer to the entire DW of finite size, i.e., we treat it as a system with one degree of freedom and neglect its non-unidimensionality in the process of the transition).

In the simplest case, setting D constant, we have the stationary solution to Eq. (6.9):

w(p) = N exp {-~(---~ ) } , (6.10)

where N is the normalized constant. It follows from this formula that the largest probability is consistent with that momentum value to which there corresponds a lower minimum of the function ~5(p). This is equivalent to the statement known as Maxwell's principle: the state of the system is determined by the global minimum of the potential function. Using this principle one can construct p(H) or v(H) seeking for the global minimum of the function ~(p) at a value of H, which varies from 0 to ec. The field dependence of the velocity will then be single-valued, i.e. there is no hysteresis, which agrees with experiment (see Chap. 4). The fields for which the values of ~ in two lower minima are comparable will have velocity jumps on the v(H) curve.

The velocities of stable stationary DW motion are found from a system of equations

O@/Op = 0, 02qS/Op 2 > 0 (6.11)

These, in the region of the nonsingle-valued function v(H) of the two minima of @, have two solutions: pl(H) and p2(H).

The Maxwell principle gives the equation to obtain the bifurcation set, i.e., the set of points in the space of governing parameters, where the transition from one local minimum to the other occurs: @(pl(H)) = 4~(p2(H). Using definition (6.7), this condition can be written as

2moll(p1 - P2) = F(p) dp , (6.12) 1

where F(p) is the total frictional force, involved in Eq. (6.6). The solution to Eq. (6.12) is given by the field corresponding to the transition to supersonic velocity on the v(H) curve. The geometric equivalence of this equation, (6.12), is the equality between two segments of areas, dashed in Fig. 6.1, which is analogous to the Maxwell rule in phase transition theory.

So, when we determine the real dependence of v on H, two approaches can be used: the principle of maximum retardation, adequate for the dynamic


P

I I I

HM H

Fig. 6.1 A scheme of constructing the dependence of v on H according to Maxwell's principle

theory, and, Maxwell's rule corresponding to an extremely strong role played by fluctuations. In order to choose which one of the two approaches ought to be used, it is necessary to solve a nonstationary equation, (6.9), to estimate the lifetime in the metastable minimum and to compare it with the time of the experiment. To this end, it is also necessary to calculate the diffusion coefficient D involved in (6.9).

D was calculated by the authors [6.3] within a specific model of randomly distributed plane defects. On the whole, the program of the analysis of nonstationary DW dynamics, with allowance for fluctuations, is far from being formulated. Moreover, it is unclear, how adequate the plane DW approxi- marion is. Experiment shows that the DW ceases to be a one-dimensional object when overcoming the sound barrier. In this situation, one of the two approaches used to describe DW dynamics in the relevant range of v(H) can be chosen only on the basis of experimental data.

The principle of maximum retardation predicts a hysteresis of the v(H) function, which is never observed in experiment. An alternative statistical approach is more preferable from the view-point of experiment. In particular, it follows from the geometrical interpretation of Maxwell's principle that the width of the near-sound region of the DW velocity steadiness (the plateau width) on the v(H) curve depends both on the resonance peak height (as for the dynamical description) and on the deflection of the curve p(H), i.e. on the initial DW mobility. With increasing mobility, the magnitude AHs (plateau width) in a statistical theory should decrease. This rule and also the absence of hysteresis on the v(H) function are consistent with experiment.

For a more exact description of the experiment, we calculate singularities on the v(H) curve near the transverse sound velocity in yt t r ium orthoferrite. Using for F(v) the formula (5.18), given above, and omitting the additve constant, we represent the Lyapunov function @(p) as:

p2


-- - - - 2mopH + Z-- -~r arctan (s - v) . (6.13) -4

The area under the resonance peak F(v) , equal to the doubled coefficient before arctan in (6.13), is independent of viscosity. Thus, the plateau width AHs , determined in a statistical theory by the Maxwell rule, is weakly depen- dent on dissipation. At the same time, the amplitude of the peak and, hence, the plateau width AHD, defined within a pure dynamic theory by making use of the principle of maximum retardation, are inversely proportional to the dissipation [6.2].

This difference is shown in Fig. 6.2. The values of A H s are found by numerical solution of the system (6.11), (6.12) for the function ~, determined by Eq. (6.13). It follows from Fig. 6.2 that with decreasing dissipation in the elastic system, resulting in the ten-fold increase of the plateau width which is given by a dynamical theory, A H s increases rather weakly. This behaviour is consistent with the fact that with lowering temperature of the sample from 300 to 4.2 K the real plateau changes insignificantly [6.6], although AHD should then increase by several orders.

200

100

AH~I, Oe

J J

J

J

T r , , , i

AHD, Oe 500 1000

Fig. 6.2 Theoretical dependence of the magnetoelastic "shelf' width AHM, calculated according to Maxwell's principle on its width AHD in the dynamical theory, for different values of the DW mobility #; curves 1, 2, 3 correspond to # = 10 8, 5 �9 10 3, and 5- 10 4 cm/s.Oe, respectively

As has already been mentioned, AHs decreases with increasing initial mobility # of the DW which corresponds experimentally to the observed phenomenon of plateau elimination in samples with high mobility, whereas AHD is independent of the DW mobility.

This is illustrated in the graphs of the A H s dependence on # given in Fig. 6.3.


AHM, Oe

200

100

i

1 2 3 4 # . 104cm/sec-Oe

Fig. 6.3 Dependence of the magnetoelastic "shelf' width Z~HM on the value of the initial mobility. 1 - AHD = 1000 Oe, 2 - AH~) = 200 Oe

As was noted in the previous section, when the corresponding conditions are satisfied, a shock wave can be formed in orthoferri te-type WFM. The shock wave excitation leads to an additional contribution to the frictional force determined by the fraction-linear function (5.34). This addition to the force of friction gives the additional term A@sw to the Lyapunov function (6.13), and, as a result, increases the plateau width A H . Using Maxwell's principle for A H s w caused by A@sw, it is easy to get:

aH w :1.

where Av = Iv:~ - s I = 2afMg/~le , t3 = (2p/moS) ( f M 2 A / ~ e ) 2, V/e is the viscosity of the elastic subsystem.

This expression is cumbersome enough and admits no analytic treatment. Its numerical analysis shows that A H s w increases sharply with decreasing viscosity of the crystal, approximately as 1 / ~ . For comparison, the A H s dependence on viscosity, ~ , is given in the same figure (Fig. 6.4) (i.e., the width anomalies caused by the usual Cherenkov mechanism). As was mentioned above, A H s depends weakly on viscosity. Thus, a t small viscosity values corresponding to low temperatures, the magnetoelastic anomaly on the dependences of v on H can be caused, primarily, by the shock wave.

6. Stabi l i ty and Probabil is t ic Description of D W Motion 95

A H

150

100

50

0.1 0.2 0.3 ~, erg.sec/cm 3

Fig . 6 .4 Dependence of the magnetoelast ic "shelf' width caused by the shock wave exci tat ion on the crystal viscosity ~e, at # = 2 . 104cm/s.Oe, s = 4.1 - 103m/s, fM3 = 3. 107erg/cm 3. Curve 1 corresponds to the nonl inear i ty coefficient (~ = 10, and curve 2 corresponds to c~ = 50. For comparison, the dependence AHM(ue) is cMculated by Maxwell 's principle for the same values of #, s, and fM3 and is presented in curve 3

06 - stability and probabilistic description of dw motion

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