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Barkhausen jumps in soft magnetic materials at low fields
M. Kléman
To cite this version:
M. Kléman. Barkhausen jumps in soft magnetic materials at low fields. Journal de Physique, 1981, 42 (9), pp.1263-1267. �10.1051/jphys:019810042090126300�. �jpa-00209317�
Barkhausen jumps in soft magnetic materials at low fields
M. Kléman
Laboratoire de Physique des Solides (*), Université Paris-Sud, Bat. 510, 91405 Orsay Cedex, France
(Reçu le 25 février 1981, accepté le 26 mai 1981)
Résumé. 2014 On propose une nouvelle théorie des sauts de Barkhausen, par laquelle on décrit le mouvement des parois à 180° par des décrochements limités par des lignes de Néel. Ces lignes peuvent s’accrocher sur les dislo- cations. On introduit deux énergies d’activation, l’une relative aux petits sauts en champ faible (énergie d’épin- glage), l’autre relative à la nucléation des lignes. On donne une expression du champ coercitif. Le traînage est expliqué par les mêmes phénomènes physiques. On compare cette théorie à une série d’expériences récentes [3, 4].
On suggère que la variation en taille des sauts de Barkhausen avec le champ reflète en partie l’ordre à courte dis-
tance des centres d’ancrage des lignes de Néel avec leur densité (ces lignes seraient flexibles aux faibles densités, quand elles sont indépendantes les unes des autres, rigides quand elles interagissent).
Abstract. 2014 A new theory of Barkhausen jumps is proposed, which depicts the movement of 180° Bloch walls as
due to kinks bounded by Néel lines. These lines can be pinned on dislocations. Two activation energies are intro- duced, one of them for small jumps in low fields and low temperatures (pinning energy), the other one for line
nucleation energy. An expression of the coercive field is given. After-effect phenomena can be explained by the same physical processes. Comparison is made with recent experiments [3, 4]. It is suggested that the variation in size of Barkhausen jumps with field partially reflects the short-range order of pinning defects, and also a change in beha- viour with the density of Néel lines (flexible when they behave independently one from the other, rigid when they interact).
Classification Physics Abstracts
b1.70 - 75.60
61.70 2013 75.60
1. Introduction. - Barkhausen jumps, i.e. irrever- sible motions of 180° Bloch walls, are well known to occur during the magnetization process, when the
magnetic field is increased beyond the initial permea-
bility range [1]. Usually they are believed to be caused by various types of obstacles (precipitates, cavities, dislocations, ...) which are able to pin the walls. Statis- tical theories relate the initial susceptibility Xi and the coercive field He to the nature and density of such pinning centres (see for example [2]) ; these theories,
which are reasonably well checked experimentally (but which anyway predict a coercive field due to dislocation pinning generally higher than the observed one) are classified as « potential theories » (the wall is
assumed to move rigidly) or « non-potential theories » (flexible walls). However they have not been used in
explaining Barkhausen jumps, and these phenomena
are indeed still not very well understood ; they possess
some important features which should be incorpo-
rated in any model : they are thermally activated
processes [3], characterized by a mean volume of spin-
(*) Associé au C.N.R.S. (L.A. 2).
flipping which varies with temperature [4, 5] ; also they play a role in after-effect [6].
Recently a series of very precise flux measurements made by Porteseil et al. [3-4] in the Rayleigh hysteresis
range, i.e. below the coercive field and in its vicinity,
have revealed a very complex behaviour of the wall motions. The authors use a sophisticated fluxmeter [4]
in which an elementary hysteresis loop as small as
1 mOe wide can be recorded with great accuracy. The
sample is a frame-shaped 3 % iron-silicon single crystal containing very few dislocations and in which the changes in magnetization are due to only one
mobile 1 80° Bloch wall. They distingûish two regimes
of displacement of the wall. When these displacements
are large (compared to the wall thickness A 0), i.e. in sufficiently high fields and high temperature, the wall
movement can be interpreted as that of a rigid object.
Barkhausenjumps seem reproducible. But on a smaller
scale or/and with smaller temperature non-reprodu-
cible Barkhausenjumps are recorded, which seem to be fairly well uncorrelated ; they are therefore attributed
to the deformation of the wall ; they are thermally
activated. Transitions between the two regimes are
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019810042090126300
1264
observed which depend on T and on the wall velocity
V.
A major difficulty in explaining the motion of a wall in a soft ferromagnet (like 3 % Si-Fe) at low fields is with the influence of strayfields. Since the anisotropy K
is so small compared to 2 nMs2 (Q = K/2 nM2 1),
any magnetization distribution at low field must avoid
magnetic poles (div M must be vanishingly small),
and wall bowing is practically forbidden except when the axis of curvature is parallel to M. Hence walls should move rigidly, which clearly requires a large
energy of friction (on obstacles) and is anyway more
or less impeded by the anchoring of 180° walls along junctions with 90° walls. Therefore some other mechanism should take place. Let us remember that an
analogous problem arises with thin walls in ferroelec- trics and in rare earth metals. Here the friction is a
Peierls-type friction, and it is usually assumed that
kinks nucleate on the wall, by thermally activated or tunnelling process, according to the case [7]. It has
also been suggested that the nucleation of the kinks is
a heterogeneous process, which takes place prefe- rentially on dislocations piercing the wall [7a, 8].
Here, although we are dealing with broad walls, we shall consider a very similar process, which is justified by the rigidity of the wall that we have discussed above.
However a major difference occurs : we shall show that the boundaries of the kinks are most probably occupied by Néel lines. Therefore we will discuss two processes taking place during Barkhausen jumps.
One of them is the activated motion of Néel lines
through the forest of dislocations piercing the wall,
very much inspired by the mechanism for dislocations, moving through the Frank network, discussed in Friedel’s book, chap. 8 [9]. It leads to a (small) coercive
field and to a logarithmic after-effect which is indeed observed [6]. The other one, which is more efficient at higher fields, concerns heterogeneous nucleation of Néel lines on dislocations piercing the wall as above
but also on dislocations parallel to it..
2. Behaviour of a Néel line in an applied field. -
The structure of a Néel line perpendicular to the magnetization in surrounding domains is given in figure 1. It is characterized by a net dipole moment perpendicular to the wall, in the core region [10, 11].
The direction of the dipole is not fixed. Suppose now
that a field H is applied, which favours one of the domains (the domain magnetized to the right, say).
The essentials of our model are in the assumption that
the wall area which is on left of the Néel line moves more easily than the area on the right, due to a rotation
of the Néel line about its axis, which is favoured by the applied field (Fig. 2). We want to estimate the energy of such a situation.
Let us first consider a segment of Néel line of length 1, strongly pinned between two points A and B
in the wall. Under an applied field H the Néel segment takes a curvature R -1 and rotates by an angle a.
Fig. 1. - Néel line (perpendicular to the figure plane) in a Bloch wall, separating two wall segments in which the magnetization
rotates with opposite chiralities. In the centre of the Néel line, there is an important component of magnetization pointing perpendi-
cular to the wall, upwards (here) or downwards with equal proba-
bilities. The Néel line is not a singular object [11].
Fig. 2. - Displacement of the wall segment on the left under the action of an applied field H ; the Néel line has rotated about its axis and the dipole is pointing in the direction of H.
’Let d o be the wall thickness. The line energy increases
by a quantity of the order of
where w is the energy per unit length of line,
and a is defined figure 2. This is counterbalanced by
the gain in wall surface energy which is twice the area
between the arc AB and its chord times H M s Jo sin a,
2 R 2 fi3
where d o sin a is the wall displacement,
and 2 R 3 2 3
thesurface swept by the arc. Clearly, in this very crude
approximation, the gain of energy is maximum for
a = n/2 : the displacement of the wall is of the order of its thickness. Hence balance of forces reads :
The radius of curvature at equilibrium is therefore
w is of the order of the exchange constant A
and probably smaller. An analytic calculation of w
is not easy in the case Q 1 ( Q = K/2 nMs2) since magnetostatic effects (which are non-local and long range) drive the distribution of M, although their
contribution to the free energy is practically negligible (this is in contrast with the case Q > 1, where an easy calculation leads to w = 8 AQ -1/2 [12]).
The numerator in equation (2) must be positive.
This requires H K/Ms, a condition which is always strongly fulfilled in our discussion. Therefore we have
This expression gives us a direct estimation of the coercive field for Néel lines moving through obstacles anchoring them very strongly. In fact, in the limit of infinite anchoring, we have to take R ~ l j2 and we
obtain
An estimation with A = 10-6 dynes, Ms = 2 x 103 G, d o = 5 X 10-6 cm, p - 104, gives H - 10-2 Oe.
This value is rather larger than those measured by
Porteseil et al. in iron (3 x 10-3 Oe) but comparable.
However we note that it is obtained in the hypothesis
of strong obstacles (and we will see that dislocations
are not of that sort) ; also it is important to remember
that Porteseil et al. observe in fact sequences of Barkhausen jumps at much smaller fields than what
they call the coercive field (3 x 10- 3 Oe).
That the interaction energy between dislocations and Néel lines is small is most evident from the fact that the sizes of a dislocation (~ b2) and of a Néel line
(~ d ô ) are very different. But we can make this more
quantitative by noticing that the interaction energy is
mostly magnetostrictive (once more we neglect stray- field energy although it plays a dominant role in fixing
the distribution of M) : the dislocation creates an
anisotropy pY of the order of AK = 3 2 À.s ub s d in its vicinity
o
Y
(this is an upper limit) ; Âs is a coefficient of magne- tostriction of the order of 10- 5 in iron. An upper limit of the anchoring energy is therefore obtained by multiplying AK by the volume of Néel line in which the dislocation stress field is effective. This is approxi- mately A02 As, where d 5 = Jo Q 1/2 is the wall thickness at the position of the line (we assume that the characte- ristic wall width here is LIs = 1t A
~A/203C0Ms2
2 nMS2 rather than- n A
since the ma netization in the line
40 = x
ft,
since the magnetization in the lineA
o K g
B
builds up against stray-field effects ; this striction of the line is not drawn in the figures). Hence
which amounts to - 1.5 x 10-12 erg in iron. Because of the simplifications of our calculation, this value compares well with the activation energy obtained in [3] for wall motion by small jumps
According to the sign of b (and of Às) the interaction is attractive or repulsive. Attractive trees are more
effective in building obstacles to the movement of a
Néel line ; they exert on the dislocations a force of the order of âUo/Ao, which opposes the force due to the
magnetic field H. By an easy extension of equation (1),
we see that a typical field He to unpin the Néel line
(at vanishing angle : rigid Néel line) is
whose order of ’ magnitude is in fact 3 x 10- 3 Oe (for AU, = 1.5 x 10-2 erg) and 3 x 10-4 Oe (if
we use the activation energy measured in (3)).
Equation (6) is akin to the equation of Orowan (9)
for yield-stress (J c’ with the following correspondance
p, in both cases, is the density of fixed dislocations crossed by the moving line. But, as in the case of Orowan’s (macro) yield-stress, it is an upper limit.
We have here micro coercive fields as we have there micro yield-stresses. The moving Néel line bows before unpinning and meets other attractive dislo- cations before H reaches Hc. The following analysis
sheds some light on this question :
Even at rest, the Néel line is not perfectly straight,
but has a zigzagging shape (with apices at attractive dislocations) whose characteristics can be estimated
as in reference [9], p. 221. It appears indeed that, since AUO is so small, the average length between pinning points on a line is very large
and the Néel lines keep quite straight. With the same
values as above we get DII - 101.5. This is in contrast to the case of dislocations moving « through the
"
forest " », where the zigzagging has a short repeat distance and is large.
D is indeed an effective mean distance between obstacles at vanishingly small fields (D = leff), which
leads to an effective coercive field He cff ~ -Il H c’c eff lef f
In fact Hceff can be better estimated by noticing that
there is at least one attractive dislocation at a distance of the Néel line - 10 do (the Néel line itself is so large
that it might contain one attractive dislocation to which it is not very efïiciently pinned, with probability
1266
Fig. 3. - Pinning of a Néel line by dislocations.
N DA 0 = 2 10-2 ; see figure 3 . This estimation
leads to
since the radius of curvature that the line must take to reach another pinning dislocation is R - I’f/20,Ao.
Hci is extremely small (- 10-’ Oe) and beyond present experimental investigations. In any case this analysis explains why very small Barkhausen jumps
can be observed below He [4, 5].
3. Mobility of Néel lines. - Let us still forget about
the possibility of nucleation of new Néel loops and apply the analysis of [9], p. 224. We have first to
investigate the free oscillations of a Néel line.
The effective mass (per unit area) mB of a Bloch wall is, according to Dôring [12]
where y is the gyrômagnetic ratio (~ 10’ s-1 Oe-1).
Wall oscillations therefore propagate with velocity
Va 1"’-1 ( (Jo/ma)1/2, where Jo is the wall surface energy [12].
Generalizing these results to a Néel line of width d o,
its effective mass per unit length is of the order of
and the characteristic velocity of oscillations is
the « Debye » frequency of a line is then
which is of the order of 1010Hz in iron.
With this value, the analog of the strain-rate equa- tion is
where r is the density of Néel loops, and
Hci is small and can be safely neglected in equation (14).
.D, in a steady state of displacement, is given by
D is of the order of Ieff for H - Hci and decreases
slowly for larger values of H. The model predicts also a logarithmic after-effect, analogous to logarithmic
creep.
Equation (13) is no longer valid for AU 0. This
happens for H - Hc (Eq. (14)) and might correspond
to the « critical velocity » observed in [4], above which the regime of small activated Barkhausen jumps is replaced by a regime of large reproducible jumps.
However we have not considered in details all the
implications of our model, and this is still an obscure
point, inasmuch as we have to take into account the
nucleation of new Néellines.
Homogeneous nucleation of loops of radius R
requires a field H = 2 M:Ao R 2 Msdoand an activation
energy of the order of
. nA 2 do
1 Th.. 1 b
l.e.
2 LB U 0
z/BUo 1 at H = H c. This is too large to be feasible,and we expect loops to nucleate on dislocations pier- cing the wall. In fact, this nucleation can be restricted
to the nucleation of two singular points of opposite strengths (figure 4 shows us indeed that a loop must
possess two singular points), whose total energy is of the order of AU, = n2 Ads i.e. 7 x 10-12 erg in iron.
This value (which here is not overestimated) is larger
than AUO and provides a second activation energy,
corresponding,presumably to the large jumps observed
in after-effects experiments near the coercive field
Hc [3]. The time of appearance of these large jumps
is of the order of a few seconds ; this is also in agree- ment, since we foresee a time of the order of
which is of the right order of magnitude.
Fig. 4. - Néel loop : a) cut perpendicular to the wall ; b) in the
wall : notice the presence of two singular points.
Note finally that dislocations parallel to the wall (in the wall) can also be responsible for the nucleation of Néel lines. We take advantage of this discussion to
point out a serious lack in all the existing discussions of the interaction of a (parallel) dislocation and a
Bloch wall [13, 14]. All these authors ignore the
influence of the chirality of the wall, but a quick inspection of the equations reveals that in most cases
there is a term in the interaction which changes sign
with chirality (in the case of the screw dislocation [13],
the total interaction changes sign). Therefore parallel
dislocations favour pinning of Néel lines. This is
important in their movement (we have not discussed
this possibility above) and in their nucleation. Vlasko- Vlasov et al. [15] have some evidence of this pheno-
menon in YIG.
4. Barkhausen jumps. - We have not yet explained
the variation with T and H of the volume vo of Bark-
hausen jumps. At very low temperatures, this volume
is the volume swept by Néel lines between attractive trees (whose density is of the order of 1/2 l’).
i.e. 2.5 x 10-10 cm3. There is probably not only
one activation energy, but a whole range (oblique and parallel dislocations, dislocation nodes). Some of these
obstacles become easier and easier to cross through
when T increases. Also, when H increases, the cur-
vature 1/R becomes large enough to help Néel lines crossing those pairs of obstacles whose distance is
larger than R. Therefore, in a way, the sequence of Barkhausen jumps reveals the short range order of obstacles.
When H approaches Hc (Eq. (6)) and when the number of Néel lines increases (by heterogeneous
nucleation as discussed above), it is no longer possible
to treat the Néel lines as independent, and they probably interact by (small) stray-fields which have
been completely neglected in our analysis. This type of interaction may probably lead to a straightening
of the lines, which then behave rigidly. Their effective coercive field is now given by equation (6). The size of
Barkhausen jumps is then dependent on obstacles
stronger than dislocations ; we can imagine that wall junctions (on which Néel lines which have moved
through the wall pile up) are such obstacles. The wall
junction would then displace suddenly when the
number of piled up Néel lines is large enough (we
owe this remark to J. Friedel).
Our model of Néel lines bounding kinks may find
some confirmation in the observations of Schôn and Büchenau [16] who saw a (static) wall displacement
from one side to the other of a Néel line in Fe 3 % Si.
Other valuable observations of Néel lines in YIG [17]
indicate that their line tension is strongly anisotropic ;
it was not possible to take into account such an effect,
in our crude calculations, as it was not possible to take
into account a possible increase of the line tension w with the angle ce of rotation of the Néel line in equa- tion (1).
Acknowledgments. - We thank Prof. J. Friedel,
Dr. J. L. Porteseil and Dr. R. Vergne for interesting
comments.
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