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A viscous friction modeling of rotational hysteresis in random anisotropy systems
B. Dieny
To cite this version:
B. Dieny. A viscous friction modeling of rotational hysteresis in random anisotropy systems. Journal de Physique, 1989, 50 (12), pp.1445-1453. �10.1051/jphys:0198900500120144500�. �jpa-00211007�
A viscous friction modeling of rotational hysteresis in
random anisotropy systems
B. Dieny
Laboratoire Louis Néel, C.N.R.S., BP 166X, 38042 Grenoble Cedex, France
(Reçu le 12 octobre 1988, révisé le 6 janvier 1989, accepté le 13 mars 1989)
Résumé. 2014 Nous présentons un modèle simple pour décrire l’évolution de l’aimantation dans des
expériences d’échantillon toumant sur des systèmes à anisotropie aléatoire. L’aimantation est divisée en domaines, chacun d’eux possédant une anisotropie uniaxiale et étant soumis à un
champ tournant. Un terme de viscosité est introduit phénoménologiquement. Ce modèle présente des transitions très brutales du nombre de rotation de l’aimantation de chaque cluster.
Les conclusions sont comparées avec les résultats expérimentaux obtenus sur un alliage amorphe
à base de terre rare.
Abstract. 2014 We present a simple model for rotating sample experiments on random anisotropy systems. The magnetization is divided into clusters, each of them having a uniaxial anisotropy and being submitted to a rotating field. A viscous friction is phenomenologically introduced. This model exhibits sharp transitions on the rotation number of the magnetization of each cluster. The conclusions are compared with experimental results obtained on an amorphous rare-earth alloy.
Classification
Physics Abstracts
75.50K - 75.60E - 75.60L - 76.90
1. Introduction.
Random anisotropy systems are characterized by a spherical distribution of local easy
magnetization axes and positive exchange interactions. The most common examples of such systems are amorphous magnetic alloys in which the cristallographic disorder induces, via crystal field effects, randomly oriented atomic anisotropies. Torque experiments on such systems are of particular interest since no preferential direction exists, a priori, on a macroscopic scale. Rotating sample experiments have recently been performed on a rare
earth amorphous alloy : DyNi [1]. In these experiments, a remanent magnetization was
established prior to rotating the sample. The behaviour of this remanent magnetization was
then studied as a function of the strength of the static field and of the rotation velocity. All the
results have been interpreted by dividing the magnetization into clusters contributing either to
a macroscopic magnetization Ml fixed with respect to the static field (weakly coercive clusters) or to a macroscopic magnetization M2 strongly bound to the sample by the anisotropy field (highly coercive clusters). Distributions of coercivity [1] and of relaxation- times [2] were associated to this partition of the magnetization.
In the present paper, we propose a simple model for the dynamics of the magnetization of
clusters submitted to a static field H in rotating sample experiments. First of all, we present
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500120144500
1446
the model itself. Secondly, we study the dynamics of the magnetization of a single cluster as a
function of two parameters : the strength of the field and the characteristic relaxation time of the cluster. Thirdly, we come back to the macroscopic scale and qualitatively compare the present results with experiments.
2. The model.
The breaking of the magnetization despite the positive exchange interactions is a common
phenomenon in random anisotropy systems. Yet, the low temperature phase has no macroscopic spontaneous magnetization and roughly consists of ferromagnetic domains (Imry
and Ma’s domains [3]) randomly oriented with respect to one another. The boundaries of these domains are not well defined surfaces since their width must be comparable to the size
of the domains themselves. In fact, statistical fluctuations of the distribution of random
anisotropy axes lead to some intrinsic inhomogeneities inside the sample : numerical
simulations [1] have illustrated that in areas where statistically the anisotropy axes happen to
be more parallel, the magnetization will be trather rigid whereas other regions where the anisotropy axes are nearly perpendicular constitute lower exchange coupling areas and
therefore will be preferential regions for cluster frontiers.
The clusters involved in low temperature hysteresis loops or torque experiments are not exactly Imry and Ma’s domains since the system is not in thermodynamic equilibrium. Their
average magnetic moment and size can be derived from the experiments [4]. For amorphous DyNi, the typical magnetic moment of a cluster determined from torque experiments has
been found to be 2 800 9 B which corresponds approximatively to 430 atoms of Dy.
In the present model, we assume that the magnetization is divided into independent clusters
of the same size and same total magnetic moment with fixed modulus M. In order to study the magnetization behaviour, we work in the reference frame of the sample and consider a
rotating field. This corresponds to the geometry of most conventional torque experiments.
Three terms are taken into account in the equation of motion :
- each cluster has a uniaxial anisotropy resulting from all the random atomic anisotropies.
Using the same expression as in Imry and Ma’s argument [3], this anisotropy amplitude D, at 0 Kelvin, should be of the order Da B/N where Da is the anisotropy on the atomic scale and N the number of magnetic atoms per cluster. The torque induced by this resultant anisotropy is TA = - D sin 2 0, 0 being the angle between the anisotropy axis and the magnetization ;
- another term accounts for the coupling with the rotating field TH = MH sin (a - 0)
where « = w t is the phase of the magnetic field ;
- a last term is introduced in order to account for frictionnal effects in the magnetization
rotation. Different experimental investigations [5-7] have reported the existence of such effects at the macroscopic scale. Thus we add a fluid frictionnal torque TF proportional and opposite to the angular g velocity y of the magnetic moment : TF = - f d o . dt Such a term is
usually introduced in the equation of motion of a magnetic wall in a metallic sample [8].
The equation of motion for the magnetic moment of a single cluster is then given by : TF + TH + TA = 0 equivalent to
with a = wt and T is the characteristic relaxation time of the cluster : 7 = f . MH (1) is a first
order non-linear equation for this sustained system. The initial condition is 0 = 0 for
a = 0 corresponding to the creation of a remanent magnetization in the direction of the field.
This equation resembles in some ways the équation of motion of a compass submitted to a static and a rotating field [9]. However the present equation is only first order and does not lead to chaotic behaviour as observed in this other problem. This equation has been numerically resolved except in some particular cases for which analytical solutions have been found.
Taking into account the wide variety of solutions of this equation and the difficulty for describing each solution, we have to introduce a reduced number of physical quantities which
are interesting for comparison with experiments. Two are of particular interest :
- the rotation number p, defined by the average value :
This rotation number can take any value between 0 and 1. If the rotation number is equal to 1, the cluster contributes to the macroscopic magnetization Ml fixed with respect to the static field, if it is equal to 0, the magnetic moment rotates with the sample so that the cluster contributes to M2. But the important point is that only these two kinds of clusters can give a macroscopic contribution. Indeed, if we plot on a circle the points given by on = 0(2 n’TT ), we
obtain only one point for p = 0 or 1. This means that if the magnetization of all clusters were
initially polarized in a narrow cone around the 0 = 0 axis, then these clusters with p = 0 or 1 respectively will together keep the same relative phase. On the other hand, for
intermediate values of p, a uniform set of points is obtained in the mapping (except for
rational values of p for which this set is discrete but symmetrical). Indeed, two clusters with
neighbouring values of p progressively lose their relative phase. As a result, the magnetization
vectors of all clusters with intermediate p give a dense set of points on the circular plot continuously changing but keeping its symmetry. Consequently, on the macroscopic scale, these clusters do not induce any magnetic flux variation parallel or transverse to the field ;
- the second interesting parameter is the average energetic loss that occurs during the
rotation. This quantity has simply to be added for all clusters for comparison with experiments. This loss is continuously compensated by the energetic supply realized by the
rotation of the field. In the present model, the loss per unit angle of sample rotation for one
cluster is given by :
The next part deals with the solutions of equation (1) for various values of the two parameters
wT and MH .
w T and
D
3. Magnetic behaviour of a single cluster.
In order to give a general view of the solutions of equation (E), we have plotted in figure 1 the phase diagram of the solutions according to their rotation number. In this diagram, two curves (numbered 1 and 2) divide the ( ( MH , D ln (w T ) plane in three areas. The two main areas
correspond to the integer values of p : 0 and 1. The third one, which exists only when
MH/ D is greater then 1 corresponds to intermediate values of p. The diagram shows that as long
as the distribution of relaxation times is wide enough, the most probable behaviour of the
1448
Fig. 1. - Phase diagram according to the rotation number in
the ( D D log (wT ) plane.
Fig. 2. - Variation of the rotation number vs. log (Cl) T ) for various values of MH : D 1.1, 1.5, 2, 3, 5, 10,
50.
magnetization of a cluster is either oscillations around a fixed direction closed to the
anisotropy axis (p = 0) or oscillations around the direction of the fields (p = 1).
More quantitatively, curve 1 is very precisely defined. Indeed, figure 2 shows the variation of the rotation number versus log ( wT ) for various values of MH . A very abrupt transition
D
occurs like at a threshold of instability between p = 1 and lower values of pu Near this
transition, a very small increase of the viscosity (or of the rotation velocity) causes the magnetization to lift out from the well of energy created by the field.
This very abrupt change of behaviour is illustrated in figure 3 : the passage from curves b to c corresponds to an increase of wT of 10- 4. Moreover, it is interesting to note that the very
rapid decrease of the rotation number is not regular at all but presents some steps for rational values of p. These steps are due to phase lock-in phenomenon which leads the magnetization
to recover the same relative phase with respect to the field, after several turns, despite an
increase of the viscosity. This phenomenon is particularly pronounced for p - 1/3 which
Fig. 3. - Characteristic set of solutions of equation (E) (a) : MH D = 0.7, wT = 0.25, p = 0 ; (b) :
MH _ 2,w T 0. 8477, p = 1 ; (c) : MH = 2,w T 0. 8478, p = 0.92 ; (d) : MH - 2, cv T = 1.05, D 2, wt = 0.8477, p = 1; (c): D 2, w’r = 0.8478, p = 0.92 ;(d): D = 2, wr = 1.05,
p = 1/3 ; (e) : MH = 2, w T = 3, p = 0. Here, we have waited three turns of the rotating field before D
=
plotting the solutions. The solutions are then calculated over sixteen turns. Figures (a) and (e) present a single curve. This means that a permanent regime, 2 7T-periodic is rapidly reached, figure (b) also
exhibits a permanent regime, but it is reached after a longer time. In the case (d), a permanent regime
exists but 6 7T-periodic. In the case (c), no periodicity is observed up to 500 turns. A jump backward
occurs regularly but never with the same phase.
1450
corresponds to the type of solution shown in figure 3 curve d. This kind of behaviour is
analogous to the lock-in phenomena observed in incommensurate phases leading to devil’s
staircase [10]. When the MH ratio becomes large, the p ( Cd T) curve reaches a limiting
D )
variation given by the solution of the equation úJ T d o - d03B1 sin (a - 0 ).
On the other hand, the curve 1 possesses a horizontal asymptote for high MH values. On
D
this asymptote, the limiting solutions of (1) can be analytically calculated and gives in permanent regime 0 = 03B1 - 03C0/2 . 03C0/2 is here the maximum possible phase shift between the field
2 2
and the magnetization of the cluster. Physically, this dephasing by 7r allows the field to exert 2
the maximum torque on the magnetization to overcome the torque of viscous origin. In other respects, the curve 1 is limited to values of MH greater than 1 by a vertical asymptote which
D
follows the numerically determined equation : wt = A MH - (MH/D-1)1 with A = 2.98 ± 0.02.
On the other hand, curve 2 is a more blurred curve. Indeed, for high values of the viscosity, figure 2 shows that the rotation number goes to 0 with a horizontal tangent. For values of MH lower than or of the order of 0.5, it is possible to determine numerically the accurate
D
value for which p reaches zero. But for higher values of MH , this determination becomes D
difficult due to numerical accuracy. Moreover, this determination is not necessary for the purpose of this study. So for comparison with experiments we define the curve 2 by the values
of w T for which p becomes smaller than 10- 2. The two curves 1 and 2 are indistinct for MH
_ 1.2 but an increasing splitting occurs for higher values of MH/D .
D g p g
D
Figures 3 gives a representative set of the different kinds of solutions of equation (1). We
refer to this figure for more details on the different solutions.
Let us now look at the dissipation that occurs during the rotation, due to a single cluster.
The variation of the normalized loss per unit angleL/ MH MH sin (a - 0 )) is plotted in
figure 4 versus wr in logarithmic scales for various D ratios.
For high values of MH (MH/D > 5 , the loss reaches an asymptotic behaviour. This
D D ) ,
asymptotic curve is characterized by two linear parts : the first one, for low viscosities, corresponds to a regime of small oscillations around a direction close to the field.
Linearisation of (1) for small (a - 0 ) ives 0 a - wt D MH 1 1+403C92 03C42 2 sin 2 (a - 03C8)
with tg g2 03C8= 2 03C8 = 2 w T. Consequently, q y the normalized dissipation is given by L - MH Cd T which is
the equation of the first asymptote in figure 4. The second linear part is observed for high
viscosities. Here, the magnetization only oscillates around the anisotropy direction. Linearis- ation of (1) for small 0 leads to
Fig. 4. - Variation of the normalized loss L sin (a - 0 ) > vs. log ( w T ) for various values of
As a result, the normalized dissipation is equal to equivalent to
1 .
h h. h... L - 1 h . - f h d f
2 w T in the high viscosity regime.L/MH= 2 /1wt is the équation of the second asymptote of
2 w T MH 2w T
figure 4. Between these two linear parts, the dissipation presents a sharp maximum with a
vertical tangent on the right side corresponding to the crossing of the critical curve oi 7- = 1 which separates the phase p - 1 from the one : p 1 (curve 1).
For low values of the MH ratio
( MH 1 ), the dissipation presents also two asymptotes :
D ( D
the first one : L = w -r ( MH 2 for low viscosities and the other one : L - 1 ,
th e lrst one:
MH
=
8 B D / J.or ow VlScosltles an t e ot er one:
MH 2 w r ’ identical to the previous case. Between these two asymptotes, a round maximum exists for
Now, for intermediate values of a curious feature is observed : a
sharp jump in the dissipation occurs corresponding to the jump of the rotation number when
one crosses the critical curve 1. The amplitude of this jump decreases and vanishes as the
rotating field strength increases. On the other hand, the insert of figure 4 shows that for these values of MH/D , the dissipation curves present some anomalies corresponding to the steps
D
1452
observed in the rotation number. When p remains equal to a constant rational value despite
an increase of the viscosity, a more abrupt decrease of the dissipation occurs with a minimum
at the threshold of instability.
4. Comparison with experiments.
In this section, we want to compare the results of this model with those obtained on DyNi [1, 2] and to discuss its validity and limitations.
The first point is that the very fact that for fields higher than 100 Oe a contribution to the total magnetization fixed with respect to the field is observed, means that the effective ratio
MH D is greater than 1. Now, from magnetic transverse relaxation experiments p [4], [] we have
found a resultant anisotropy barrier acting on a cluster of 160 K at 4.7 K and the Zeeman energy for a cluster in the lowest experimental field of 100 Oe can be estimated : MH = 2 800 03BCB B X 100 Oe ~ 20 K. This gives a MH ratio of only 0.13. This discrepancy may
D
be mostly attributed to the absence of any thermal phenomena in the model. The cluster rotation must involve thermally activated nucleation propagation phenomena as in hysteresis loop. It is well known that these processes considerably lower the coercivity in real samples.
However the very abrupt jump observed from p = 1 to lower values of p for
w T = 1 can justify the hypothesis of reference 2 according to which the relaxation time distribution of the clusters could be deduced from the rotation velocity dependence of Ml (Ml is the modulus of the magnetization fixed with respect to the field). Indeed, a small
decrease of the rotation period d(Log Tu) leads the clusters, the relaxation time of which is around 03C4R , to lift out from the well of energy treated by the field and get into the area of
2 Tr
intermediate rotation number values where they do not contribute any more to the
macroscopic magnetization. In consequence, a decrease of Mi occurs proportionally to the
number of these clusters. Thus, except for ratio MH close to 1, the distribution of D
d log M1 log 03C4 should effectively be given by the normalized derivative dlogMl .
d log 03C4R
Concerning the dissipation, let us firstly compare the order of magnitude obtained for the maximum of loss in the field range used : 100 Oe, 1.5 kOe. If we assume that all clusters have the same relaxation time and that the rotation velocity corresponds to the maximum of loss :
sin ( a - 0 ) ) = 1 then we should find a total loss per gramme and per unit angle of rotation of 150 x 103 erg/g in 1 kOe to be compared to 45 x 103 erg/g found experimentally.
Considering that the width of the relaxation-time distribution is about two decades [2, 4],
all clusters cannot be exactly at their maximum of dissipation for the same rotation velocity.
As a result, the experimental loss should be lower than this maximum value of 150 x 103 erg/g by a factor of the order of 5, and the shape of the maximum must be rounder than for a single cluster. These conclusions agree with the experimental results. However, if
we consider now the decreasing of the dissipation on both sides of this maximum, the model predicts two simple asymptotic behaviours for one cluster : for MH > 5 and 1,
D
L = w T x MH which gives g L proportional to TR- 1 ; for w T > 1, L = 1 2 w TMH implying L proportional to TR. Such dependences should lead to a more abrupt decrease of the
dissipation on both sides of the maximum than that observed experimentally. This discrepancy may be partly ascribed to the width of the relaxation-time distribution. We may also invoke the possibility of a distribution of effective anisotropy on each cluster leading to a
mixing of curves corresponding to different MH/D ratios. A significant contribution of clusters
D
with MH/D close to 1 should lead to a wide broadening of the maximum and to a slower
decrease of loss for low rotation velocity.
In conclusion, we have studied a model of individual magnetic moments with uniaxial
anisotropy, submitted to a rotating field in a viscous medium. This model exhibits sharp
transitions on the rotation number. The two main situations encountered are either rotation with the field or small oscillations around the anisotropy axis. This feature can qualitatively justify the interpretation of rotating sample experiments made on a random anisotropy system : amorphous DyNi. However a more complete model is needed to describe correctly
the dissipation and to obtain the right order of magnitude for the force which binds the
magnetization to the sample. In particular, thermal activation, nucleation and propagation phenomena should be taken into account.
Acknowledgments.
1 wish to thank C. Godrèche (CEA Saclay) and P. Collet (SPT Ecole Polytechnique) for their
lectures in the summer school of Beg-Rohu (June 1988) and B. Barbara for valuable discussions about the physics of the model.
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