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Intrinsic coercive field in pseudobinary cubic intermetallic compounds. I: DyxY1−xAl2 and Dy1−δAl2
370
Journal
INTRINSIC COERCIVE FIELD IN PSEUDOBINARY COMPOUNDS. I: Dy,Y,_,Al, AND Dy,_bAl, J.I. ARNAUDAS,
of Magnetism
and Magnetic
Materials 61 (1986) 370-380 North-Holland, Amsterdam
CUBIC INTERMETALLIC
A. DEL MORAL
Institute de Ciencia de Materiales (ICMA) and Departamento Universidad de Zaragoza, 5ooO9-Zaragoza, Spain
de Electricidad y Magnetism0
(Far&ad
de Ciencias), C.S.I.C.
and
and J.S. ABELL Department Received
of Metallurgy
21 February
and Materials,
University of Birmingham,
1986; in revised form 31 March
Birmingham BI5 2TT, UK
1986
The thermal variation of the coercive field in pseudodobinary Laves phase compounds Dy,Y,-,Al, (0.10 < x ~1.0) and Dy, _s Al, (6 = 0.025 and 0.048) has been measured from 3.7 K up to the Curie temperatures. It turns out that the coercive field has two contributions: one from the intrinsic.pinning of narrow domain walls in the Peierls crystal potential wells of the anisotropy energy; this contribution dominates at low temperatures and decays exponentially with the narrow domain wall width. The other comes from the pinning of the domain walls by deflecs, being explained by the Kersten theory of critical domain wall bowing. The domain walls seem very narrow in these materials, the widths estimated from the coercive field being between two and three atomic spacings.
1. Introduction In the early 1970s Barbara et al. [l], Egami and Graham [2], Zijlstra [3], Van den Broek and Zijlstra [4], and Hilzinger and Kronmiiller [5] recognized the existence of narrow domain walls in magnetic materials where the anisotropy energy is comparable to the exchange energy. For a 1800 narrow domain wall in a uniaxial material the position where the center of the wall lies in an atomic plane has a higher energy than when it lies between two planes [3], the Peierls energy barrier separating both positions pinning the narrow domain wall at any point in the lattice and giving rise to an intrinsic coercive field. This energy barrier, AE, was initially calculated by van den Broek and Zijlstra [4] and Egami [6] for uniaxial rare earth (RE) materials; van den Broek and Zijlstra used a discrete approximation and the result is that AE scales rapidly with the ratio K/A between the anisotropy (K) and exchange (A)
energies. The structure and width of the narrow domain walls was initially calculated by Barbara [7] for a simple cubic RE intermetallic and by Egami and Graham [2] for Dy and Tb metals, finding narrow domain walls of about 7 atomic layers. The first successful attempt to relate the intrinsic coercive field, Hi’, to the width of the narrow domain wall was made by Hilzinger and Kronmiiller [5,8] for hexagonal ferromagnets, namely SmCo,. They used both discrete and continuous inter-spin angle approximations attaining the same result: they found H,i - exp( - rS/D), where S is the domain wall width and D the interplanar spacing within the wall. In this way they were able to explain the low-temperature coercive field in SmCo,. Since then a number of theoretical papers about the intrinsic coercive field in SmCo, have appeared, mainly from Kronmiiller and coworkers [9,10] and Craik et al. [11,12]. Narrow domain walls have been recognized in a
0304-8853/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
J. I. Amauahs et al. / Coercive field in pseudobinaty
number of RE materials, e.g. in non-collinear ferromagnets such as Dy, Al 2 [13] and DyNi [14], and in hexagonal pseudobinaries RE(Co,_,M,), (M = Cu, Ni, Al) [15-191. Although some relevant work has been done on magnetization processes in pseudobinaries of Laves phase intermetallics, mainly in RE (Fe,Al,_,), and RE (Co, Al,_,), by Oesterreicher et al. [20-221 and Grijssinger et al. [23], and in Dy (CoNi), by Taylor et al. [24], a thorough study of the coercive field in Laves phase intermetallics is lacking. In particular for the REAl, ones, if we consider that these systems are those of the most thoroughly RE intermetallics magnetically studied to date [25]. Therefore we have carried out a systematic study of the coercive field, remanence, magnetic relaxation and initial reversible susceptibility in a series of RE,RE;_, Al, pseudobinaries, and also in the binaries REAl,. We will restrict this paper to the coercive field in the Dy,Y, _-x Al, series, together with Dy, _ 6Al 2 compounds slightly deficient in Dy; for them (100) is the easy direction of magnetization [26]. In these compounds it is to be expected that substitution of Dy by Y will reduce the exchange interactions, decreasing in principle the width of the narrow domain walls [27].
2. Theory of coercive field in REAl, cubic intermetallics The above-mentioned exponential decrease of the intrinsic coercive field with the domain wall width found by Hilzinger and Kronmiiller in hexagonal materials [5,8,10] does not seem to hold for cubic systems, where the magnetocrystalline anisotropy energy is basically different. Therefore we have developed a calculation essentially similar to those authors, but specialized to ferromagnets of Laves phase structure. In addition, we will perform a continuous inter-spin angle calculation, considering the success of the continuous approximation of Hilzinger and Kronmtiller [8]. A comparison of our results for the domain wall structure with the semi-discrete approximation of Barbara [7] as applied to DyAl, by Bowden et al. [28] is also fully satisfactory. The domain wall of
compounds. I
371
lowest energy in cubic Fe, according to Lilley’s [29] calculations, is a 90” one parallel to the (001) plane, and therefore such a domain wall is most probable. Therefore we will restrict the calculation of the intrinsic coercive field to such a type of domain wall, although the strength of the calculated intrinsic coercive field for other kinds of domain walls is roughly the same. 2.1. Domain
wall structure and energy
For the Laves phase structure (fig. 1) the exchange energy per unit area of domain wall can be written as
(2.1) where we only consider the interaction between nearest neighbours; $, is the angle of the spin S, within the nth atomic plane, with the [loo] axis, and the summation extends over all planes parallel to (001). # is the exchange integral and S = (g - 1) J, J being the total angular momentum and g the Land& factor of the ion; a is the lattice constant. The anisotropy energy can be written as yk=
g
n=-CC
$ K, sin2#, cos’#,,,
Fig. 1. (001) domain the spins at different ing).
(2.2)
wall for Laves phases structure showing atomic layers (D is the interlayer spac-
312
J.I. Amaudas
et al. /
Coercive field in pseudobinary compounds. I
where K, is the first cubic anisotropy constant. Transforming now the discrete angle 4, to a continuous one \c,(z), z being the coordinate along the normal to the domain wall, the total energy y = y,. + yk becomes
the average domain wall energy at equilibrium, y=y[i
ln(chg)+$],
(2.8)
which for domain walls of three or more atomic planes can be approximated by
(2.9) where the exchange constant A =#(S’)/a and .$= JZ& (S2) is the thermal average value of the square of the ionic spin. In equilibrium 6y = 0 and the standard Euler variational calculation gives, for the domain wall, the differential equation i2 sin*+ cos2# = $I’* - $
[ t/f4 + 2q!+” - #rr2] , (2.4)
keeping terms up to fourth order. A perturbation solution of (2.4), starting from the solution of the second-order equation, gives the following domain wall equation: $(z)=arctanexp
&z+&[‘([z--y
th.$z)
1.
This solution holds even for domain walls as narrow as about three atomic planes; for &a < 2m, what happens for domain walls wider than about three atomic planes, (2.5) transforms into the simpler solution for the second-order approximation of (2.4): I/J(Z) = arc tan exp Ez
or
sin 2$=
The value m differs from 7 by less than 5%, even for a domain wall of S = 20, where D = a/4 is the interplanar spacing (see fig. 1). Therefore the complete solution (2.5), with domain wall energy given by (2.8), is very near to the second-order one, as given by eqs. (2.6) and (2.9); this justifies the use of a second-order approximation in order to evaluate the intrinsic Peierls energy barrier in section 2.2. It is important to ascertain how much our continuous approximation deviates from a discrete treatment; in fig. 2 we show the domain wall structure as calculated from (2.5) compared with the discrete treatment result of Bowden et al. [28], and where we can see that the deviation is very small. 2.2. Calculation of the intrinsic coercive field Displacing the center of the domain wall from the position of minimum anisotropy energy (be-
l/ch ,$‘z, (2.6)
which corresponds to a domain wall width given by
(2.7) A similar continuous
calculation
[8] gives, for
Fig. 2. Variation of the spin angle (4) along a normal coordidiscrete calculanate z for a 90” (001) domain wall. ( -) tion (ref. [28]); (- - -) continuous angle approximation (see eq. (2.5)).
J.I. Arnaudas et al. / Coercive field in pseudobinary
tween two atomic planes), and displacing further the domain wall from this position a small distance z [S] the inter-spin angle becomes, from (2.6): +,, = f arc sin { l/ch[
t( nD - D/2
- z)] } . (2.10)
Substitution of (2.10) into (2.1) and (2.2) gives, for the displaced domain wall energy: v(z)=?
= +
(2.11)
E (Y;(z)+Y,ex), n= --m
t2
c
r=fl +5
[I-((
l+ch[2.$(nD-D/2-z)
(at + ar)
C
ah” =
e2aihr/D,
(2.13)
with the use of eqs. (2.12a), (2.12b), we for the Fourier coefficients the expressions:
t!i(-l)h/m e2;;2;‘D dy,
(2.14a)
--oo
.._4A _--
ah
(2.16a)
,-r=/ID.
(2.16b)
(2.17)
(2.12b)
h=-cc
where obtain
e-T2’ED,
- D/2 - z)
being respectively the anisotropy and exchange energy contributions. y(z) must be periodic within the crystal lattice and can conveniently be Fourier expanded [S], i.e. =
(2.14b) is extremely difficult to A numerical calculation gives:
Therefore the variation of domain wall energy for a small displacement z from its equilibrium position becomes, from (2.16a), (2.16b):
rD])
+E’D])-1)1’2],
y(z)
a: = 4.876874
a; = -2+
(ch [rD + ch[2E(nD
X
(2.15)
a sh( vr’h/s$D) ’
(2.12a)
“=z4ch2+D-D,2-z]’ u??(z)=
2r2h
a:=(-1)“A
the coefficients for 1h 1 > 1 being negligible compared with ay. For 6 > D the predominant coefficient at can be adequately approximated by
where a2
It is immediate to check that a? and at are the average domain wall equilibrium energy U,, and yk, as given by (2.8). The integral (2.14a) can be exactly solved giving, for the Fourier anisotropy coefficients,
but the integral solve analytically.
E Y”(Z) n=-CC
313
compoundr. I
1
a 50
where the Peierls energy barrier Ay = ( - 19.507 + 167r2){
is
e-r(s’o).
(2.18)
The result (2.17), in agreement with the one obtained by Hilzinger and Kronmiiller [5,8] for hexagonal uniaxial materials, shows that the domain wall energy varies periodically with its lattice position. The energy barrier Ay, according to (2.18) decreases exponentially with the domain wall width S; although most of this energy is anisotropy energy, the barrier is reduced by = 12% by the negative exchange contribution. The intrinsic coercive field, which wipes out the local maxima of the Peierls potential, will be given by
(2.14b) where we have performed the change of variables, y=(nD-D/2-z)t and w=2[(nD-z)+[D, respectively.
and from (2.17) and (2.18) for a 90” (001) domain
314
J.I. Arnaudas et al. / Coercive field in pseudobinary
wall is obtained
that (2.19)
where W = A/a*, A4, is the spontaneous magnetization, and C a constant depending on the type and orientation of the wall, and equal to 1.23 X lo3 for a 90° (001) domain wall. Other types of domain walls are possible, but with higher energies, and are therefore less probable; so a 90° (110) domain wall has an energy = 1.72&,0(O01). A 180” (001) domain wall needs magnetostrictive distortions to be stabilized [30], but considering it to be formed by two 90” domain walls H, becomes only double than for a 90” domain wall. Therefore to analyze the coercivity of the present compounds in terms of 90” (001) domain walls is quite realistic. The intrinsic coercive field can be further reduced from the value given by (2.19) by the homogeneous thermal activation of kinks [6], displacing the domain wall a distance D over a circular region, which is energetically more favourable than the displacement of the whole domain wall. The kink expands irreversibly above a certain critical radius and, as calculated by Kutterer et al. [9], this effect reduces H,, by a factor
iI
compouncis. I
can be basically reduced to a certain power of the reduced magnetization, mB, with /3 varying roughly between -2.5 and 22.5 [31]. Very high values of p are unphysical, and negative values can be discarded except for low temperatures; we have tested in the isomorphous series of intermetallics Tb,Gd, _xA12 many existing theories of extrinsic coercive field [32], in particular those of NCel [33] and Vicena [34,35] related to random internal stresses, finding either a poor fitting of the experiment or unphysical results. The best fitting to the high temperature variation of the coercive field (see figs. 4a-f below) in the present series was obtained with the simple Kersten theory [36,37]. In this model the domain wall, pinned by lines of defects, bows until a critical radius, from where it moves irreversibly under a coercive field given by H,e” = &p’/*, s
(2.22)
where k’ = 2 and 2fi for 180“ and 90” domain walls, respectively, and p is the density of defects per unit area of domain wall. This coercive field has a temperature dependence H,““(T) = H,““(0)[m(T)]4.5.
(2:23)
2 l/2
k(T)=
l-
[
f
i
P
(2.20)
-f, P
where Tp = B(ya*/Ka), with B a constant, is the temperature where k(T) is reduced to 40% of its value at 0 K [9]. Therefore the full expression for the intrinsic coercive field becomes: Hi(T)
= C k(T);
e-R(S/D).
(2.21)
s
2.3. Extrinsic
coercive field
The pinning of domain walls by, in principle, all kinds of defects (internal stresses, grain inhomogeneities and precipitates, boundaries, phase boundaries, dislocations, etc.) will contribute to a coercive field that we will call “extrinsic”. All theories of an extrinsic coercive field give for the coercive field a temperature variation which
3. Experimental
details and samples
3.1. Samples The samples were prepared by melting the constituents several times in an argon arc furnace. These were Dy and Y of 99.9% purity (Rare Earth Products Ltd.) and Al of 99.999% (Koch Light Labs.); samples of Dy,Y,_,Al, with x = 1, 0.95, 0.90, 0.80, 0.60, 0.50, 0.35, 0.20 and 0.10 were prepared. Discs of about 6 mm in diameter and 1 mm in thickness were cut from the boules by spark erosion in order to perform the coercive field measurements. All the compounds were checked by the X-ray powder technique, the measured lattice parameters being in good agreement with the literature [38]; they vary linearly with x in good agreement with Vegard’s law. No second phases were found,
J.I. Arnaudas et al. / Coercive field in pseudobinary compounds. I
- 10 pm. Similarly, in unetched samples, tally tiny lamellar and globular precipitates of black colour, with typical distances among them between - 1 and 10 pm, were observed by optical microscopy. These figures will become relevant below for the analysis of the extrinsic coercive field.
by this analysis, within 5% confidence. For the study of a coercive field it is essential to fully characterize the samples and therefore metallographic analyses were undertaken, using optical microscopy and SEM in the back scattered electron mode. A full account of this study will be published elsewhere, and we will only summarize here the main results. In the compounds with x = 1, 0.8, 0.6 and ($5 a second phase rich in Al was observed, although of a composition very near to the Laves phases; on the opposite side, for x = 0.35 and 0.2 a second phase was observed deficient in Al, but this phase too contained a third one even richer in RE. X-ray microprobe and EDAX analysis agree with such deficiencies. Using contrast by channelling in the BSE mode, it was possible to neatly observe grains that for x = 0.8, for example, range between 150 and 450 pm. Observation by optical microscopy with oblique incidence and etching appropriately the surfaces gave an average grain size of - 400 pm for the whole series. Etch-pitch of dislocations were observable by attacking the surfaces with dilute hydrochloric acid, the average distance between pits being typi-
M(emu/gr)
315
3.2, Coercive field measurement The coercive field was measured by applying a field of 2.04 T to the disc samples, enough to technically saturate the magnetization and reducing it until obtaining zero magnetization. Due to the presence of magnetic relaxation, with times of - 100 s at the coercive field and 4.2 K (and higher for lower fields and higher temperatures), it was necessary to ensure that the zero magnetization observed was not produced by the relaxation effect at a magnetic field smaller than the true coercive field. So the applied field was rapidly (in - 20 s) reduced from the maximum value to values each time more negative, obtaining the graphs shown in fig. 3: the coercive field was then calculated from the cut of the lines with the field axis.
1
/
38K 72K P
/
Fig. 3. Magnetization isotherms for Dy,,sY,.,Al,
compound used to determine coercive fields from the intercept with the H field axis.
316
J.I. Arnaudas et al. / Coercive field in pseudobinary
The slopes of the lines are l/N, N being the demagnetizing factor, but this fact obviously does not affect the coercive field values.
T
500
I
I
1
,
I
1
compounds. I
4. Experimental
results, analysis and discussion
A powerful method to ascertain the origin of the coercive field, as became apparent in the dis1
r
I
1
,
b
-3 I
a
I
I
1
,
(
C
i
Fig. 4. (a) Thermal variation of the measured coercive field, H,, for Dy,Y,_,Al, series for different compositions, x(O). (- - -) calculated contribution of the intrinsic coercive field with TP = 10.7 K (from eq. (2.21)); (- -. -) calculated extrinsic coercive field contribution from eq. (2.23)); () calculated overall coercive field (from eq. (4.1)). (b) As (a) with TP = 10 K. (c) As (a) with 7” = 9.9 K. (d) As (a) with TP = 4.7 K. (e) As (a) with TP = 4.3 K. (f) As (a) with TP = 2.8 K.
J.I. Arnauah
et al. / Coercive field in pseudobinary
cussion in section 3, is to study its evolution with temperature. We should expect a large difference between the wavelengths of the intrinsic and extrinsic potential variations and therefore the coercive field should be given by the summation of both contributions [9], H,(T)
= H,(T)
+ HP(T),
(4.1)
as given by expressions (2.21) and (2.23). In figs. 4a-f we show the thermal variation of H, for the present series of compounds; for x G 0.2 the coercive field was zero, within the experimental error (= f 0.5 Oe). In fig. 5 we show also the temperature dependence of the logarithm of H,, where linear regions are observed at low temperatures. Therefore this plot allows us to determine by extrapolation the coercive field at 0 K, H,(O), and also the Curie points (H, = 0) with better accuracy than from fig. 4. We should note that for some compounds (x = 1, 0.9, 0.5 and 0.35) the coercive field was measured both in the samples as cast and after annealing at 800°C during 76 h in high vacuum (- 5 X 10P6 Torr), not observing any significant difference in H,. This is in good agreement with a similar observation of the nondependence on annealing of the radio frequency domain wall enhancement factor in NMR experiments for this series [26]. In figs. 4a-f we show the fitting of the experimental temperature variation of H, for different compounds, using the theory previously developed (expressions (4.1), (2.21) and (2.23)); as may be seen, the fitting is, in general, rather good. As is observed, the contribution of the intrinsic coercive field is predominant at low temperatures, going down rapidly; at high temperatures the origin of the coercive field is mostly extrinsic, and this field remains low and virtually constant at low temperatures. The extrinsic coercive field was calculated from (2.23) using the magnetization vs. temperature results obtained for the present compounds [39]. The intrinsic coercive field was calculated, according to (2.21), in the following way. From spin wave theory the exchange function W(T) is given by [40]
W(T) =
S,(T),
(4.2)
compounds. I
371
where z is the most probable number of Dy” ions nearest neighbours of any one and calculated using the binomial distribution law [26]. The domain wall width was calculated according to (2.7), which predicts a thermal variation, 6(T) = S(0) [m(T)]-4.5. In figs. 4a-f we can see the calculated thermal variation of H,‘(T). There are two adjustable parameters, C and rP. C was adjusted to the same value, 251.3, for the whole series, in qualitative agreement with the theory in subsections 2.2 and 2.3. The values of TP that give the best fitting at high temperatures are included in figs. 4a-f. r’ increases with Dy content, in qualitative agreement with the theory of subsection 2.2 which predicts a proportionality to 7. Relevant information which can be extracted from the present measurements is the domain wall widths; from (2.21) the average numbers of spins within the domain wall at 0 K is given by
where H,‘(O) = H,(O) - HP(O). In fig. 6 we show the compositional dependence of Hi(O), together with n(0). As can be seen, the variation of n(0) with x is very weak, but enough to produce huge variations in H,!(O). From the estimated domain wall widths, these are very narrow, between two and three atomic spacings (the calculated value for a 90” (001) domain wall in DyAl, is = 3.5, taking K,(O) = 36 K/ion, [26] and W(0) = 2.9 K/ion)). The increase of Hi between x = 0.8 and 0.6 is in good agreement with the weakening of the radio frequency NMR enhancement factor n (- l/H,‘), as observed by Barbara and Berthier [26]. They obtained the value H,(DyAl,) = 1.5 kOe at 1.4 K, higher than our 0 K value, i.e. 400 Oe. As pointed out by these authors the higher value could be a consequence of the dynamic technique used. We can also estimate the compositional dependence of the first anisotropy constant K,(O), using the relation obtained from (2.7): K,(O) = 16n2W(0)/n2(0).
(4.4)
The value obtained for DyAl, is 67 K/ion, almost double the value of 36 K/ion obtained by Barbara and Berthier [26]; however, there is good
J.I. Arnaudas ef al. / Coercive field in pseudobinary 3-
compounds I
I
I
a z 0 4
I
1
PLY
600-
A’2
I
500-
9 f
LOO-
w 6
I-
d’
fJ 300-
!
3
I ‘a.-&
8 $zoo-
*
a
-c*+x-
I
-2
0. $ 5
i
$ 100 -
D~xYl-xA12
!
-1
I Ill
0
n
A
02
I 0 6
I
0.4
I 08
X,lat %I
YAI,
D-
1
I 1
DYAIz
Fig. 6. Compositional dependence (x is the atomic 5%of Dy) of the deduced intrinsic coercive field in Dy,Y, _xA12 series (0 and -. -. -). Also shown the compositional dependence of the determined domain wall widths, Aa, at 0 K (- - -).
I-
O
I
I
10
20
I
I
,
b,
TEMPERATURE
I
60
LO (K)
Fig. 5. Logarithmic plot of the thermal variation of the coercive field, H,, for DyxY,_,A12 intermetallic compounds of different compositions (the continuous lines are guides for the eye).
agreement considering the pinning of the 90” (110) domain wall (D = aJ?J/4), where K,(O) becomes 33.5 K/ion. The compositional dependence of K,(O), normalized to the value of 36 K/ion for x = 1, is shown in fig. 7; K,(O) scales linearly with x, as expected from a single-ion origin for anisotropy, up to x = 0.7 and then remains constant. Now, in order to fit adequately the thermal dependence of the magnetization along [loo] for these compounds (mainly at high x concentrations), it happened to be necessary to introduce negative quadrupolar coupling interactions [ 381, which would in turn provide a negative contribution to
K,, thus explaining the invariance of K,(O) at high Dy concentrations. The complicated compositional dependence of Hz(O) could in principle be explained by a weakening of the exchange constant (A) with the initial substitution of Dy by Y, up to = 0.6, producing an initial increase of H,!(O), followed by a decrease produced by a delicate interplay of both exchange (A) and anisotropy (K,) energies below x = 0.6. From the determined extrinsic coercive field and using eq. (2.22) we can also estimate the
1
LO
I
I
DyxYl-x Alz
‘“L 0
02
YAl2
OL
06 x (at
% I
I
I
0
06
000
1 DYAIz
Fig. 7. Compositional dependence of the deduced cubic anisotropy constant K, at 0 K in the Dy;Y, -*Al, series (see text for details).
J.I. Arnaudm
et al. / Coercive field in pseudobinaty
average distance between pinning centers considering that p = 1/Z2, if we admit that the defects are disposed in a cubic array of lattice constant 1. The estimated value is then I = 1 pm, which is in fair agreement with the observed spacings between etch-pits of dislocations and between precipitates in these samples (see subsection 3.1). Finally, we have also investigated the effect on the coercive field produced by a slight deficiency in Dy, preparing compounds of formula Dy, _,r Al, with 6 = 0.025 and 0.048. These compounds have the Laves phase structure with lattice constants 7.8348 and 7.8375 A, respectively. In fig. 8 is shown the thermal variation of a coercive field for both compounds together with the fitted intrinsic
DY,+,A$
0
20
60
60
TEMP&T"RE,[K)
Fig. 8. Thermal variation of the measured coercive field, H,, for the non-stoichiometric compounds Dy,_s Al, (A, 6 = 0.025; 0, 8 = 0.048). (- - - and - - -) calculated intrinsic coercive field for 6 = 0.025 and 0.048, respectively; (- .-.-) calculated extrinsic coercive field contributions; ( -) calculated overall coercive field (eq. (4.1)).
compounds. I
319
and extrinsic contributions. Now the domain wall widths S(0) are around 2.7 atomic spacings, slightly higher than for the Dy,Y, _XA12 compounds with equal Dy content (see fig. 6). This may be a consequence of the weakening of the anisotropy energy by the introduction of vacancies at the RE sublattice, as expected if we consider that the Curie temperatures are very close to the Dy,Y, _-xAl 2 corresponding compounds (T, = 59 and 55 K for 6 = 0.025 and 0.048, respectively).
5. Conclusions It has been shown in the series of cubic Laves phases Dy,Y,_,Al, (x > 0.2) and Dy,_, Al, (S = 0.025, 0.048) that the coercive field at low enough temperatures is mainly intrinsic, produced by the homogeneous pinning of narrow domain walls in the Peierls crystal potential wells. The energy barrier separating the minima has been shown to decrease exponentially with the domain wall width, in agreement with results in hexagonal structure materials [5,8,9]. From the estimated domain wall widths, they are very narrow, between two and three atomic spacings. The dependence of the intrinsic coercive field with the Dy content can be adequately explained by the interplay between the exchange and anisotropy energies, as modified by homogeneous Yttrium substitutions. On the other hand, the coercive field contribution by defects (extrinsic coercive field) present in these materials has been well explained by Kersten’s [36,37] theory of critical bowing and irreversible displacement of the domain walls pinned by the defects. The estimated average separation between pinning centers as obtained from the extrinsic coercive field is very small (about 1 pm) in these intermetallics.
Acknowledgements We are grateful to the Spanish CAICYT for the provision of financial support under grant 385/81. J.I. Amaudas wishes to acknowledge the support of the British Council for time spent at the Department of Metallurgy and Materials, University
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J.I. Amaudas et al. / Coercive field in pseudobinary compounds. I
of Birmingham, and we are grateful to Dr. D.W. Jones for the facilities provided.
References [l] B. Barbara, C. Belle, R. Lemaire and D. Paccard, J. de Phys. 32 (1971) Cl-299. [2] T. Egami and C.D. Graham, J. Appl. Phys. 42 (1971) 1299. [3] H. Zijlstra, IEEE Trans. Magn. MAG-6 (1970) 179. [4] J.J. van den Broek and H. Zijlstra, IEEE Trans. Mag. MAG-7 (1971) 226. [5] H.R. Hilzinger and H. Kronmiiller, Phys. Stat. Sol. (b) 54 (1972) 593. [6] T. Egami, Phys. Stat. Sol. (a)19 (1973) 747. [7] B. Barbara, J. de Phys. 34 (1973) 1039. [8] H. Hilzinger and H. Kronmliller, Phys. Stat. Sol. (b)59 (1973) 71. [9] R. Kiitterer, H.R. Hilzinger and H. Kronmiiller, J. Magn. Magn. Mat. 4 (1977) 1. [lo] H. Kronmtiller, J. Magn. Magn. Mat. 7 (1978) 341. [ll] D.J. Craik and E. Hill, Phys. Lett. A 48 (1974) 157. [12] D.J. Craik, E.W. Hill and A.J. Harrison, IEEE Trans. Magn. MAG-11 (1975) 1379. [13] B. Barbara, C. Belle, R. Lemaire and D. Paccard, CR. Acad. Sci. (Paris) 271 (1970) B-880. [14] D. Gignoux and J.S. Shah, Solid State Commun. 11 (1972) 1709. [15] M. F%hnle and H. Oesterreicher, Phys. Rev. B27 (1983) 5586. [16] H. Maeda, J. Appl. Phys. 50 (1979) 2346. [17] M. Fahnle and H. Oesterreicher, J. Magn. Magn. Mat. 38 (1983) 331.
[18] K.H. Buschow and M. Brouha, J. Appl. Phys. 47 (1976) 1653. [19] A.I. Ermolenko, A.V. Korolyov and A.F. Rozhda, IEEE Trans. Magn. MAG-13 (1977) 1339. [20] H. Oesterreicher, J. Appl. Phys. 42 (1971) 5137. [21] H. Oesterreicher and R. Pitts, J. Appl. Phys. 43 (1972) 5174. [22] H. Oesterreicher, J. Phys. Chem. Solids 34 (1973) 1267. [23] R. Grossinger, W. Steiner and K. Krec, J. Magn. Magn. Mat. 2 (1976) 196. [24] K.N.R. Taylor, D. Melville and G.J. Primavesi, J. Phys. F 2 (1972) 584. [25] K.H.J. Buschow, Ferromagnetic Materials, vol. 1, ed. E.P. Wohlfarth (North-Holland, Amsterdam, 1980) p. 299. [26] B. Barbara and Y. Berthier, Physica 86-88B (1977) 1385. [27] G.J. Bowden and R.K. Ray, J. Phys. F 8 (1978) 533. [28] G.J. Bowden, H.M. Cadogan, K.R. Doolan, P.J. Martinson and J.M. Pope, J. Phys. F 12 (1982) 363. [29] B.A. Lilley, Phil. Mag. 41 (1950) 792. [30] C. Kittel and J.K. Galt, Solid State Phys. 3 (1956) 437. [31] J.I. Amaudas, Ph.D. Thesis, University of Zaragoza (1985). [32] J.I. Amaudas, A. de1 Moral, P.A. Algarabel and J.S. Abell, J. Magn. Magn. Mat. 61 (1986) 381. [33] L. NCel, Ann. Univ. Grenoble 22 (1946) 299. [34] F. Vicena, Czech. J. Phys. 4 (1954) 419. [35] F. Vicena, Czech. J. Phys. 5 (1955) 11. [36] M. Kersten, Z. Angew. Phys. 7 (1956) 313. [37] M. Kersten, Z. Angew. Phys. 8 (1956) 382. [38] K.H.J. Buschow, Rep. Prog. Phys. 42 (1979) 1373. [39] J.I. Amaudas and A. de1 Moral, to be published. [40] W. Daring, Handbuch der Physik, vol. 18, no. 2 (Springer-Verlag, Berlin, 1966).
Journal
INTRINSIC COERCIVE FIELD IN PSEUDOBINARY COMPOUNDS. I: Dy,Y,_,Al, AND Dy,_bAl, J.I. ARNAUDAS,
of Magnetism
and Magnetic
Materials 61 (1986) 370-380 North-Holland, Amsterdam
CUBIC INTERMETALLIC
A. DEL MORAL
Institute de Ciencia de Materiales (ICMA) and Departamento Universidad de Zaragoza, 5ooO9-Zaragoza, Spain
de Electricidad y Magnetism0
(Far&ad
de Ciencias), C.S.I.C.
and
and J.S. ABELL Department Received
of Metallurgy
21 February
and Materials,
University of Birmingham,
1986; in revised form 31 March
Birmingham BI5 2TT, UK
1986
The thermal variation of the coercive field in pseudodobinary Laves phase compounds Dy,Y,-,Al, (0.10 < x ~1.0) and Dy, _s Al, (6 = 0.025 and 0.048) has been measured from 3.7 K up to the Curie temperatures. It turns out that the coercive field has two contributions: one from the intrinsic.pinning of narrow domain walls in the Peierls crystal potential wells of the anisotropy energy; this contribution dominates at low temperatures and decays exponentially with the narrow domain wall width. The other comes from the pinning of the domain walls by deflecs, being explained by the Kersten theory of critical domain wall bowing. The domain walls seem very narrow in these materials, the widths estimated from the coercive field being between two and three atomic spacings.
1. Introduction In the early 1970s Barbara et al. [l], Egami and Graham [2], Zijlstra [3], Van den Broek and Zijlstra [4], and Hilzinger and Kronmiiller [5] recognized the existence of narrow domain walls in magnetic materials where the anisotropy energy is comparable to the exchange energy. For a 1800 narrow domain wall in a uniaxial material the position where the center of the wall lies in an atomic plane has a higher energy than when it lies between two planes [3], the Peierls energy barrier separating both positions pinning the narrow domain wall at any point in the lattice and giving rise to an intrinsic coercive field. This energy barrier, AE, was initially calculated by van den Broek and Zijlstra [4] and Egami [6] for uniaxial rare earth (RE) materials; van den Broek and Zijlstra used a discrete approximation and the result is that AE scales rapidly with the ratio K/A between the anisotropy (K) and exchange (A)
energies. The structure and width of the narrow domain walls was initially calculated by Barbara [7] for a simple cubic RE intermetallic and by Egami and Graham [2] for Dy and Tb metals, finding narrow domain walls of about 7 atomic layers. The first successful attempt to relate the intrinsic coercive field, Hi’, to the width of the narrow domain wall was made by Hilzinger and Kronmiiller [5,8] for hexagonal ferromagnets, namely SmCo,. They used both discrete and continuous inter-spin angle approximations attaining the same result: they found H,i - exp( - rS/D), where S is the domain wall width and D the interplanar spacing within the wall. In this way they were able to explain the low-temperature coercive field in SmCo,. Since then a number of theoretical papers about the intrinsic coercive field in SmCo, have appeared, mainly from Kronmiiller and coworkers [9,10] and Craik et al. [11,12]. Narrow domain walls have been recognized in a
0304-8853/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
J. I. Amauahs et al. / Coercive field in pseudobinaty
number of RE materials, e.g. in non-collinear ferromagnets such as Dy, Al 2 [13] and DyNi [14], and in hexagonal pseudobinaries RE(Co,_,M,), (M = Cu, Ni, Al) [15-191. Although some relevant work has been done on magnetization processes in pseudobinaries of Laves phase intermetallics, mainly in RE (Fe,Al,_,), and RE (Co, Al,_,), by Oesterreicher et al. [20-221 and Grijssinger et al. [23], and in Dy (CoNi), by Taylor et al. [24], a thorough study of the coercive field in Laves phase intermetallics is lacking. In particular for the REAl, ones, if we consider that these systems are those of the most thoroughly RE intermetallics magnetically studied to date [25]. Therefore we have carried out a systematic study of the coercive field, remanence, magnetic relaxation and initial reversible susceptibility in a series of RE,RE;_, Al, pseudobinaries, and also in the binaries REAl,. We will restrict this paper to the coercive field in the Dy,Y, _-x Al, series, together with Dy, _ 6Al 2 compounds slightly deficient in Dy; for them (100) is the easy direction of magnetization [26]. In these compounds it is to be expected that substitution of Dy by Y will reduce the exchange interactions, decreasing in principle the width of the narrow domain walls [27].
2. Theory of coercive field in REAl, cubic intermetallics The above-mentioned exponential decrease of the intrinsic coercive field with the domain wall width found by Hilzinger and Kronmiiller in hexagonal materials [5,8,10] does not seem to hold for cubic systems, where the magnetocrystalline anisotropy energy is basically different. Therefore we have developed a calculation essentially similar to those authors, but specialized to ferromagnets of Laves phase structure. In addition, we will perform a continuous inter-spin angle calculation, considering the success of the continuous approximation of Hilzinger and Kronmtiller [8]. A comparison of our results for the domain wall structure with the semi-discrete approximation of Barbara [7] as applied to DyAl, by Bowden et al. [28] is also fully satisfactory. The domain wall of
compounds. I
371
lowest energy in cubic Fe, according to Lilley’s [29] calculations, is a 90” one parallel to the (001) plane, and therefore such a domain wall is most probable. Therefore we will restrict the calculation of the intrinsic coercive field to such a type of domain wall, although the strength of the calculated intrinsic coercive field for other kinds of domain walls is roughly the same. 2.1. Domain
wall structure and energy
For the Laves phase structure (fig. 1) the exchange energy per unit area of domain wall can be written as
(2.1) where we only consider the interaction between nearest neighbours; $, is the angle of the spin S, within the nth atomic plane, with the [loo] axis, and the summation extends over all planes parallel to (001). # is the exchange integral and S = (g - 1) J, J being the total angular momentum and g the Land& factor of the ion; a is the lattice constant. The anisotropy energy can be written as yk=
g
n=-CC
$ K, sin2#, cos’#,,,
Fig. 1. (001) domain the spins at different ing).
(2.2)
wall for Laves phases structure showing atomic layers (D is the interlayer spac-
312
J.I. Amaudas
et al. /
Coercive field in pseudobinary compounds. I
where K, is the first cubic anisotropy constant. Transforming now the discrete angle 4, to a continuous one \c,(z), z being the coordinate along the normal to the domain wall, the total energy y = y,. + yk becomes
the average domain wall energy at equilibrium, y=y[i
ln(chg)+$],
(2.8)
which for domain walls of three or more atomic planes can be approximated by
(2.9) where the exchange constant A =#(S’)/a and .$= JZ& (S2) is the thermal average value of the square of the ionic spin. In equilibrium 6y = 0 and the standard Euler variational calculation gives, for the domain wall, the differential equation i2 sin*+ cos2# = $I’* - $
[ t/f4 + 2q!+” - #rr2] , (2.4)
keeping terms up to fourth order. A perturbation solution of (2.4), starting from the solution of the second-order equation, gives the following domain wall equation: $(z)=arctanexp
&z+&[‘([z--y
th.$z)
1.
This solution holds even for domain walls as narrow as about three atomic planes; for &a < 2m, what happens for domain walls wider than about three atomic planes, (2.5) transforms into the simpler solution for the second-order approximation of (2.4): I/J(Z) = arc tan exp Ez
or
sin 2$=
The value m differs from 7 by less than 5%, even for a domain wall of S = 20, where D = a/4 is the interplanar spacing (see fig. 1). Therefore the complete solution (2.5), with domain wall energy given by (2.8), is very near to the second-order one, as given by eqs. (2.6) and (2.9); this justifies the use of a second-order approximation in order to evaluate the intrinsic Peierls energy barrier in section 2.2. It is important to ascertain how much our continuous approximation deviates from a discrete treatment; in fig. 2 we show the domain wall structure as calculated from (2.5) compared with the discrete treatment result of Bowden et al. [28], and where we can see that the deviation is very small. 2.2. Calculation of the intrinsic coercive field Displacing the center of the domain wall from the position of minimum anisotropy energy (be-
l/ch ,$‘z, (2.6)
which corresponds to a domain wall width given by
(2.7) A similar continuous
calculation
[8] gives, for
Fig. 2. Variation of the spin angle (4) along a normal coordidiscrete calculanate z for a 90” (001) domain wall. ( -) tion (ref. [28]); (- - -) continuous angle approximation (see eq. (2.5)).
J.I. Arnaudas et al. / Coercive field in pseudobinary
tween two atomic planes), and displacing further the domain wall from this position a small distance z [S] the inter-spin angle becomes, from (2.6): +,, = f arc sin { l/ch[
t( nD - D/2
- z)] } . (2.10)
Substitution of (2.10) into (2.1) and (2.2) gives, for the displaced domain wall energy: v(z)=?
= +
(2.11)
E (Y;(z)+Y,ex
t2
c
r=fl +5
[I-((
l+ch[2.$(nD-D/2-z)
(at + ar)
C
ah” =
e2aihr/D,
(2.13)
with the use of eqs. (2.12a), (2.12b), we for the Fourier coefficients the expressions:
t!i(-l)h/m e2;;2;‘D dy,
(2.14a)
--oo
.._4A _--
ah
(2.16a)
,-r=/ID.
(2.16b)
(2.17)
(2.12b)
h=-cc
where obtain
e-T2’ED,
- D/2 - z)
being respectively the anisotropy and exchange energy contributions. y(z) must be periodic within the crystal lattice and can conveniently be Fourier expanded [S], i.e. =
(2.14b) is extremely difficult to A numerical calculation gives:
Therefore the variation of domain wall energy for a small displacement z from its equilibrium position becomes, from (2.16a), (2.16b):
rD])
+E’D])-1)1’2],
y(z)
a: = 4.876874
a; = -2+
(ch [rD + ch[2E(nD
X
(2.15)
a sh( vr’h/s$D) ’
(2.12a)
“=z4ch2+D-D,2-z]’ u??(z)=
2r2h
a:=(-1)“A
the coefficients for 1h 1 > 1 being negligible compared with ay. For 6 > D the predominant coefficient at can be adequately approximated by
where a2
It is immediate to check that a? and at are the average domain wall equilibrium energy U,, and yk, as given by (2.8). The integral (2.14a) can be exactly solved giving, for the Fourier anisotropy coefficients,
but the integral solve analytically.
E Y”(Z) n=-CC
313
compoundr. I
1
a 50
where the Peierls energy barrier Ay = ( - 19.507 + 167r2){
is
e-r(s’o).
(2.18)
The result (2.17), in agreement with the one obtained by Hilzinger and Kronmiiller [5,8] for hexagonal uniaxial materials, shows that the domain wall energy varies periodically with its lattice position. The energy barrier Ay, according to (2.18) decreases exponentially with the domain wall width S; although most of this energy is anisotropy energy, the barrier is reduced by = 12% by the negative exchange contribution. The intrinsic coercive field, which wipes out the local maxima of the Peierls potential, will be given by
(2.14b) where we have performed the change of variables, y=(nD-D/2-z)t and w=2[(nD-z)+[D, respectively.
and from (2.17) and (2.18) for a 90” (001) domain
314
J.I. Arnaudas et al. / Coercive field in pseudobinary
wall is obtained
that (2.19)
where W = A/a*, A4, is the spontaneous magnetization, and C a constant depending on the type and orientation of the wall, and equal to 1.23 X lo3 for a 90° (001) domain wall. Other types of domain walls are possible, but with higher energies, and are therefore less probable; so a 90° (110) domain wall has an energy = 1.72&,0(O01). A 180” (001) domain wall needs magnetostrictive distortions to be stabilized [30], but considering it to be formed by two 90” domain walls H, becomes only double than for a 90” domain wall. Therefore to analyze the coercivity of the present compounds in terms of 90” (001) domain walls is quite realistic. The intrinsic coercive field can be further reduced from the value given by (2.19) by the homogeneous thermal activation of kinks [6], displacing the domain wall a distance D over a circular region, which is energetically more favourable than the displacement of the whole domain wall. The kink expands irreversibly above a certain critical radius and, as calculated by Kutterer et al. [9], this effect reduces H,, by a factor
iI
compouncis. I
can be basically reduced to a certain power of the reduced magnetization, mB, with /3 varying roughly between -2.5 and 22.5 [31]. Very high values of p are unphysical, and negative values can be discarded except for low temperatures; we have tested in the isomorphous series of intermetallics Tb,Gd, _xA12 many existing theories of extrinsic coercive field [32], in particular those of NCel [33] and Vicena [34,35] related to random internal stresses, finding either a poor fitting of the experiment or unphysical results. The best fitting to the high temperature variation of the coercive field (see figs. 4a-f below) in the present series was obtained with the simple Kersten theory [36,37]. In this model the domain wall, pinned by lines of defects, bows until a critical radius, from where it moves irreversibly under a coercive field given by H,e” = &p’/*, s
(2.22)
where k’ = 2 and 2fi for 180“ and 90” domain walls, respectively, and p is the density of defects per unit area of domain wall. This coercive field has a temperature dependence H,““(T) = H,““(0)[m(T)]4.5.
(2:23)
2 l/2
k(T)=
l-
[
f
i
P
(2.20)
-f, P
where Tp = B(ya*/Ka), with B a constant, is the temperature where k(T) is reduced to 40% of its value at 0 K [9]. Therefore the full expression for the intrinsic coercive field becomes: Hi(T)
= C k(T);
e-R(S/D).
(2.21)
s
2.3. Extrinsic
coercive field
The pinning of domain walls by, in principle, all kinds of defects (internal stresses, grain inhomogeneities and precipitates, boundaries, phase boundaries, dislocations, etc.) will contribute to a coercive field that we will call “extrinsic”. All theories of an extrinsic coercive field give for the coercive field a temperature variation which
3. Experimental
details and samples
3.1. Samples The samples were prepared by melting the constituents several times in an argon arc furnace. These were Dy and Y of 99.9% purity (Rare Earth Products Ltd.) and Al of 99.999% (Koch Light Labs.); samples of Dy,Y,_,Al, with x = 1, 0.95, 0.90, 0.80, 0.60, 0.50, 0.35, 0.20 and 0.10 were prepared. Discs of about 6 mm in diameter and 1 mm in thickness were cut from the boules by spark erosion in order to perform the coercive field measurements. All the compounds were checked by the X-ray powder technique, the measured lattice parameters being in good agreement with the literature [38]; they vary linearly with x in good agreement with Vegard’s law. No second phases were found,
J.I. Arnaudas et al. / Coercive field in pseudobinary compounds. I
- 10 pm. Similarly, in unetched samples, tally tiny lamellar and globular precipitates of black colour, with typical distances among them between - 1 and 10 pm, were observed by optical microscopy. These figures will become relevant below for the analysis of the extrinsic coercive field.
by this analysis, within 5% confidence. For the study of a coercive field it is essential to fully characterize the samples and therefore metallographic analyses were undertaken, using optical microscopy and SEM in the back scattered electron mode. A full account of this study will be published elsewhere, and we will only summarize here the main results. In the compounds with x = 1, 0.8, 0.6 and ($5 a second phase rich in Al was observed, although of a composition very near to the Laves phases; on the opposite side, for x = 0.35 and 0.2 a second phase was observed deficient in Al, but this phase too contained a third one even richer in RE. X-ray microprobe and EDAX analysis agree with such deficiencies. Using contrast by channelling in the BSE mode, it was possible to neatly observe grains that for x = 0.8, for example, range between 150 and 450 pm. Observation by optical microscopy with oblique incidence and etching appropriately the surfaces gave an average grain size of - 400 pm for the whole series. Etch-pitch of dislocations were observable by attacking the surfaces with dilute hydrochloric acid, the average distance between pits being typi-
M(emu/gr)
315
3.2, Coercive field measurement The coercive field was measured by applying a field of 2.04 T to the disc samples, enough to technically saturate the magnetization and reducing it until obtaining zero magnetization. Due to the presence of magnetic relaxation, with times of - 100 s at the coercive field and 4.2 K (and higher for lower fields and higher temperatures), it was necessary to ensure that the zero magnetization observed was not produced by the relaxation effect at a magnetic field smaller than the true coercive field. So the applied field was rapidly (in - 20 s) reduced from the maximum value to values each time more negative, obtaining the graphs shown in fig. 3: the coercive field was then calculated from the cut of the lines with the field axis.
1
/
38K 72K P
/
Fig. 3. Magnetization isotherms for Dy,,sY,.,Al,
compound used to determine coercive fields from the intercept with the H field axis.
316
J.I. Arnaudas et al. / Coercive field in pseudobinary
The slopes of the lines are l/N, N being the demagnetizing factor, but this fact obviously does not affect the coercive field values.
T
500
I
I
1
,
I
1
compounds. I
4. Experimental
results, analysis and discussion
A powerful method to ascertain the origin of the coercive field, as became apparent in the dis1
r
I
1
,
b
-3 I
a
I
I
1
,
(
C
i
Fig. 4. (a) Thermal variation of the measured coercive field, H,, for Dy,Y,_,Al, series for different compositions, x(O). (- - -) calculated contribution of the intrinsic coercive field with TP = 10.7 K (from eq. (2.21)); (- -. -) calculated extrinsic coercive field contribution from eq. (2.23)); () calculated overall coercive field (from eq. (4.1)). (b) As (a) with TP = 10 K. (c) As (a) with 7” = 9.9 K. (d) As (a) with TP = 4.7 K. (e) As (a) with TP = 4.3 K. (f) As (a) with TP = 2.8 K.
J.I. Arnauah
et al. / Coercive field in pseudobinary
cussion in section 3, is to study its evolution with temperature. We should expect a large difference between the wavelengths of the intrinsic and extrinsic potential variations and therefore the coercive field should be given by the summation of both contributions [9], H,(T)
= H,(T)
+ HP(T),
(4.1)
as given by expressions (2.21) and (2.23). In figs. 4a-f we show the thermal variation of H, for the present series of compounds; for x G 0.2 the coercive field was zero, within the experimental error (= f 0.5 Oe). In fig. 5 we show also the temperature dependence of the logarithm of H,, where linear regions are observed at low temperatures. Therefore this plot allows us to determine by extrapolation the coercive field at 0 K, H,(O), and also the Curie points (H, = 0) with better accuracy than from fig. 4. We should note that for some compounds (x = 1, 0.9, 0.5 and 0.35) the coercive field was measured both in the samples as cast and after annealing at 800°C during 76 h in high vacuum (- 5 X 10P6 Torr), not observing any significant difference in H,. This is in good agreement with a similar observation of the nondependence on annealing of the radio frequency domain wall enhancement factor in NMR experiments for this series [26]. In figs. 4a-f we show the fitting of the experimental temperature variation of H, for different compounds, using the theory previously developed (expressions (4.1), (2.21) and (2.23)); as may be seen, the fitting is, in general, rather good. As is observed, the contribution of the intrinsic coercive field is predominant at low temperatures, going down rapidly; at high temperatures the origin of the coercive field is mostly extrinsic, and this field remains low and virtually constant at low temperatures. The extrinsic coercive field was calculated from (2.23) using the magnetization vs. temperature results obtained for the present compounds [39]. The intrinsic coercive field was calculated, according to (2.21), in the following way. From spin wave theory the exchange function W(T) is given by [40]
W(T) =
S,(T),
(4.2)
compounds. I
371
where z is the most probable number of Dy” ions nearest neighbours of any one and calculated using the binomial distribution law [26]. The domain wall width was calculated according to (2.7), which predicts a thermal variation, 6(T) = S(0) [m(T)]-4.5. In figs. 4a-f we can see the calculated thermal variation of H,‘(T). There are two adjustable parameters, C and rP. C was adjusted to the same value, 251.3, for the whole series, in qualitative agreement with the theory in subsections 2.2 and 2.3. The values of TP that give the best fitting at high temperatures are included in figs. 4a-f. r’ increases with Dy content, in qualitative agreement with the theory of subsection 2.2 which predicts a proportionality to 7. Relevant information which can be extracted from the present measurements is the domain wall widths; from (2.21) the average numbers of spins within the domain wall at 0 K is given by
where H,‘(O) = H,(O) - HP(O). In fig. 6 we show the compositional dependence of Hi(O), together with n(0). As can be seen, the variation of n(0) with x is very weak, but enough to produce huge variations in H,!(O). From the estimated domain wall widths, these are very narrow, between two and three atomic spacings (the calculated value for a 90” (001) domain wall in DyAl, is = 3.5, taking K,(O) = 36 K/ion, [26] and W(0) = 2.9 K/ion)). The increase of Hi between x = 0.8 and 0.6 is in good agreement with the weakening of the radio frequency NMR enhancement factor n (- l/H,‘), as observed by Barbara and Berthier [26]. They obtained the value H,(DyAl,) = 1.5 kOe at 1.4 K, higher than our 0 K value, i.e. 400 Oe. As pointed out by these authors the higher value could be a consequence of the dynamic technique used. We can also estimate the compositional dependence of the first anisotropy constant K,(O), using the relation obtained from (2.7): K,(O) = 16n2W(0)/n2(0).
(4.4)
The value obtained for DyAl, is 67 K/ion, almost double the value of 36 K/ion obtained by Barbara and Berthier [26]; however, there is good
J.I. Arnaudas ef al. / Coercive field in pseudobinary 3-
compounds I
I
I
a z 0 4
I
1
PLY
600-
A’2
I
500-
9 f
LOO-
w 6
I-
d’
fJ 300-
!
3
I ‘a.-&
8 $zoo-
*
a
-c*+x-
I
-2
0. $ 5
i
$ 100 -
D~xYl-xA12
!
-1
I Ill
0
n
A
02
I 0 6
I
0.4
I 08
X,lat %I
YAI,
D-
1
I 1
DYAIz
Fig. 6. Compositional dependence (x is the atomic 5%of Dy) of the deduced intrinsic coercive field in Dy,Y, _xA12 series (0 and -. -. -). Also shown the compositional dependence of the determined domain wall widths, Aa, at 0 K (- - -).
I-
O
I
I
10
20
I
I
,
b,
TEMPERATURE
I
60
LO (K)
Fig. 5. Logarithmic plot of the thermal variation of the coercive field, H,, for DyxY,_,A12 intermetallic compounds of different compositions (the continuous lines are guides for the eye).
agreement considering the pinning of the 90” (110) domain wall (D = aJ?J/4), where K,(O) becomes 33.5 K/ion. The compositional dependence of K,(O), normalized to the value of 36 K/ion for x = 1, is shown in fig. 7; K,(O) scales linearly with x, as expected from a single-ion origin for anisotropy, up to x = 0.7 and then remains constant. Now, in order to fit adequately the thermal dependence of the magnetization along [loo] for these compounds (mainly at high x concentrations), it happened to be necessary to introduce negative quadrupolar coupling interactions [ 381, which would in turn provide a negative contribution to
K,, thus explaining the invariance of K,(O) at high Dy concentrations. The complicated compositional dependence of Hz(O) could in principle be explained by a weakening of the exchange constant (A) with the initial substitution of Dy by Y, up to = 0.6, producing an initial increase of H,!(O), followed by a decrease produced by a delicate interplay of both exchange (A) and anisotropy (K,) energies below x = 0.6. From the determined extrinsic coercive field and using eq. (2.22) we can also estimate the
1
LO
I
I
DyxYl-x Alz
‘“L 0
02
YAl2
OL
06 x (at
% I
I
I
0
06
000
1 DYAIz
Fig. 7. Compositional dependence of the deduced cubic anisotropy constant K, at 0 K in the Dy;Y, -*Al, series (see text for details).
J.I. Arnaudm
et al. / Coercive field in pseudobinaty
average distance between pinning centers considering that p = 1/Z2, if we admit that the defects are disposed in a cubic array of lattice constant 1. The estimated value is then I = 1 pm, which is in fair agreement with the observed spacings between etch-pits of dislocations and between precipitates in these samples (see subsection 3.1). Finally, we have also investigated the effect on the coercive field produced by a slight deficiency in Dy, preparing compounds of formula Dy, _,r Al, with 6 = 0.025 and 0.048. These compounds have the Laves phase structure with lattice constants 7.8348 and 7.8375 A, respectively. In fig. 8 is shown the thermal variation of a coercive field for both compounds together with the fitted intrinsic
DY,+,A$
0
20
60
60
TEMP&T"RE,[K)
Fig. 8. Thermal variation of the measured coercive field, H,, for the non-stoichiometric compounds Dy,_s Al, (A, 6 = 0.025; 0, 8 = 0.048). (- - - and - - -) calculated intrinsic coercive field for 6 = 0.025 and 0.048, respectively; (- .-.-) calculated extrinsic coercive field contributions; ( -) calculated overall coercive field (eq. (4.1)).
compounds. I
319
and extrinsic contributions. Now the domain wall widths S(0) are around 2.7 atomic spacings, slightly higher than for the Dy,Y, _XA12 compounds with equal Dy content (see fig. 6). This may be a consequence of the weakening of the anisotropy energy by the introduction of vacancies at the RE sublattice, as expected if we consider that the Curie temperatures are very close to the Dy,Y, _-xAl 2 corresponding compounds (T, = 59 and 55 K for 6 = 0.025 and 0.048, respectively).
5. Conclusions It has been shown in the series of cubic Laves phases Dy,Y,_,Al, (x > 0.2) and Dy,_, Al, (S = 0.025, 0.048) that the coercive field at low enough temperatures is mainly intrinsic, produced by the homogeneous pinning of narrow domain walls in the Peierls crystal potential wells. The energy barrier separating the minima has been shown to decrease exponentially with the domain wall width, in agreement with results in hexagonal structure materials [5,8,9]. From the estimated domain wall widths, they are very narrow, between two and three atomic spacings. The dependence of the intrinsic coercive field with the Dy content can be adequately explained by the interplay between the exchange and anisotropy energies, as modified by homogeneous Yttrium substitutions. On the other hand, the coercive field contribution by defects (extrinsic coercive field) present in these materials has been well explained by Kersten’s [36,37] theory of critical bowing and irreversible displacement of the domain walls pinned by the defects. The estimated average separation between pinning centers as obtained from the extrinsic coercive field is very small (about 1 pm) in these intermetallics.
Acknowledgements We are grateful to the Spanish CAICYT for the provision of financial support under grant 385/81. J.I. Amaudas wishes to acknowledge the support of the British Council for time spent at the Department of Metallurgy and Materials, University
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of Birmingham, and we are grateful to Dr. D.W. Jones for the facilities provided.
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