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Magnetic properties of amorphous ferromagnetic alloys
Journal of Magnetism and Magnetic Materials 13 (1979) 53-70 0 North-Holland Publishing Company
MAGNETIC PROPERTIES H. KRONMULLER,
OF AMORPHOUS FERROMAGNETIC
ALLOYS
M. FAHNLE, M. DOMANN, H. GRIMM, R. GRIMM, B. GROCER
Institut fiir Physik am Max-Planck-Institut fiir Metallforschung,
Stuttgart, Fed. Rep. Germany
Received 11 April 1979
For the application of amorphous, ferromagnetic alloys as soft magnetic materials a detailed knowledge of the domain and thermodynamic properties is required. A review of the present state of the research work will be given. It is shown that the displacement of domain walls is described fairly well by the conventional potential theory developed previously for crystalline materials. Within the framework of the theory of micromagnetism the effect of quenching stresses on the domain structure as well as of short range stresses on the magnetic microstructure are discussed in detail,
structure, magnetization processes, the magnetic microstructure
1. Introduction
talline materials. An important feature of amorphous alloys is, furthermore, the absence of any grain or phase boundaries which are known to act as strong pinning centres for dws in crystalline materials. In amorphous alloys therefore one of the main sources of pinning centres is eliminated. Nevertheless the initial permeability of amorphous alloys in general is a factor of 5 smaller than that of crystalline materials. However, for finite magnetic fields of H w 20 mOe (0.02 A cm-‘) the permeability is comparable with that of crystalline materials [l-3]. On the other hand the coercive fields are in general found to be smaller than those of crystalline soft magnetic materials. The origin of this rather peculiar behaviour must be attributed to properties of the domain structure as well as of the microstructure. Extensive investigations have been performed with respect to intrinsic properties of amorphous alloys; e.g., the spontaneous magnetization and the Curie temperature have been investigated for many alloys (see review articles by Egami and Graham [4], Luborsky et al. [3], and Hilzinger et al. [I]). Only a few investigations, however, are known which deal with an interpretation of the characteristic properties of the hysteresis loop. For a solution of these problems investigations in the following fields are required: (1) domain patterns; (2) magnetization processes; (3) magnetic anisotropy energy;
Amorphous, ferromagnetic alloys have turned out to be a promising new soft magnetic material with attractive mechanical and magnetic material properties. Some of the outstanding properties are the following: (1) small coercive fields; (2) large permeability; (3) small magnetic losses; (4) small magnetic anisotropy; (5) large critical shear stress. As is well known in crystalline materials lattice imperfections act as rather effective sources for the destruction of excellent soft magnetic properties, This undesired effect is in general due to strong magnetoelastic interactions between domain walls (dws) and the internal stresses of lattice defects. Therefore even small plastic deformations of crystalline magnetic materials lower the permeability considerably. Generally a careful annealing treatment is required in order to reduce these pinning effects of microstructural defects. In addition the magnetostriction is reduced to approximately zero by developing appropriate alloys. Well known examples of these alloys are Deltamax, Permalloy and Supermalloy. As plastic deformations are rather destructive for soft magnetic materials amorphous alloys with their large critical shear stresses have significant advantages in comparison with crys53
54
H. Kronmiiller et al. /Magnetic
properties of amorphous ferromagnetic
(4) interaction between microstructure and domain walls; (5) characterization of pinning centres in amorphous alloys. The aim of this paper is a concise review of the above mentioned topics. For a study of domain structures, macroscopic magnetization processes and anisotropic energies we have applied the method of the magneto-optical Kerr effect, and for a study of the microstructure we have investigated the Rayleigh region and analyzed measurements of the law of approach to ferromagnetic saturation. In addition these latter investigations give important information on the spin wave dispersion relation and the temperature dependence of the spontaneous magnetization. Our experimental results will be discussed within the framework of the theory of micromagnetism as already developed for amorphous materials [S-7]. It will be shown that the displacement of dws may be described successfully within the framework of potential theory.
2. Domain structures in amorphous alloys 2.1. General remarks For the study of domain structures in amorphous alloys the classical Bitter technique [8-l l] as well as the magnetooptical Kerr effect has been successfully applied [12-161. The first of these techniques may be applied to a study of surface domain structures of the as-quenched specimens whereas the latter technique generally requires a careful surface polishing. Therefore the Bitter technique is useful for a study of the effect of surface roughnesses on the dws whereas the Kerr technique is more appropriate for a study of the change of the dws under the influence of an applied magnetic field. As is well known domain structures are the result of a complicated energy balance between magnetostatic stray field energies, anisotropic magnetic energy terms, and the exchange energy usually contained in the domain wall energy. In soft magnetic materials the stray field energy &L,-,M~is much larger than anisotropy energies and the domain wall energy. The dws therefore is submitted to the condition of stray field
alloys
free spin configurations. This rather restrictive condition imposes the following for the dws: (1) disappearance of volume charges div M,(r) = 0, (2) disappearance
(1) of surface charges
n(r) - M,(r) = 0,
(2)
where n(r) corresponds to the unit vector perpendicular to the surface element at position r. Eq. (2) is fulfilled if the spontaneous magnetization at the surface is aligned parallel to the surface. Due to (1) and (2) the geometrical arrangement of domains is exclusively determined by magnetic anisotropy, the dw energy, and the shape of the specimen. 2.2. Anisotropy energies The domain patterns of amorphous alloys are determined by the following three different types of anisotropy energies: fie shape anisotropy. In order to reduce the magnetostatic stray field energy the spontaneous magnetization throughout the specimen tries to align parallel to the ribbon’s surface. Due to the smaller demagnetization field for a magnetization direction parallel to the ribbon’s axis the spontaneous magnetization prefers to be aligned parallel to the ribbon. The magnetoelastic coupling energy. In a material with non-vanishing isotropic magnetostriction, X,, internal stresses, 0, give rise to a magnetostrictive interaction energy
where cii corresponds to the ith component of the diagonalized stress tensor and -yi denotes the direction cosine of MS with respect to the ith coordinate axis. Structure anisotropy. The absence of long range order averages out those energy terms like the magnetic anisotropy in crystals. Nevertheless due to temperature gradients or magnetic fields occurring during the quenching process there may exist gradients in the distribution of metalloid atoms. Furthermore some atomic pair ordering of short range type (e.g., Fe-Ni pairs) may be induced due to gradients in the atomic distribution function.
H. Kronmiiller et al. /Magnetic properties of amorphous ferromagnetic alloys
55
In the following we present domain patterns where the above mentioned anisotropy energies play a dominant role. 2.3. Domain structures Our magneto-optical Kerr effect investigations were performed on amorphous NiFePB- and CoFeSiB-alloys as produced by the spin-quenching technique. The specimens were prepared by grinding with diamond paste and were then electrolytically polished in a solution of 4 nitric acid and $ methanol. After this treatment the specimens generally had a thickness of about 20 pm. In order to increase the optical contrast the polished specimens were evaporated with a A/4-i& of ZnS [ 171. Characteristic domain patterns observed in spinquenched alloys are shown in figs. l-6. 2.3.1. iQFe4oP14B6 (Metglas 2826) [13,1.5] In the general view given in fig. 1 for the domain structure for zero applied field three different types of domain patterns may be observed: (i) wide, wavy laminae with a width of 25 pm. In some regions these wide laminae show preferred orientations parallel or perpendicular to the ribbon axis. (ii) Patches of rather narrow, sometimes circular or zig-zag laminae with a domain width of ~3-5 pm. These patches have diameters of about 150-200 pm
Fig. 1. Domain structure of an asquenched narrow zig-zag laminae (H = 0).
Fe40Ni40P14B6-ribbon
Fig. 2. Domain structure of narrow laminae compressive stresses (H = 0, Metglas).
in a region
of
and in general are embraced by wide laminae (see fig. 2). (iii) Starlike domain patterns where wide laminae are flowing out from one centre (see fig. 3). After an annealing treatment for 20 h at 473 K the wide laminae were oriented perpendicular to the ribbon axis (rolling direction) and the number of domain patches was considerably reduced (see fig. 4). Under an applied magnetic field the domain
at zero-applied
field with regions
of wide starlike
domains
and
56
H. Kronmiiller
Fig. 3: Starlike domain pattern stresses (H = 0, Metglas).
et al. /Magnetic
properties of amorphous ferromagnetic
alloys
in a region of dilatational
patterns show characteristic changes. Wide domains vanish at such low fields as H = 10 Oe (800 A m-l) (see fig. 5) whereas the domain patches vanish at 20 Oe (1600 A m-l) if the magnetic field is applied perpendicular to the narrow laminae of the patches (fig. 6). In a magnetic field applied parallel to the narrow
Fig. 4. Domain
structure
after an annealing
Fig. 5. Change of the domain field H = 800 A m-t applied laminae (Metglas).
treatment
structure under a magnetic perpendicular to the narrow
of 20 h at 473 K (H = 0, Metglas).
H. KronmtiIler et al. /Magnetic properties of amorphous ferromagnetic alloys
51
laminae these laminae vanish at the rather high fields of 180 Oe (14 400 A m-r). 2.3.2. FexoBzo (Allied Chemical 2605) [16] As shown in fig. 7 in FesoBzo similar domain patterns are observed as in the case of Ni4eFe4ePr4B6, For an applied field of 22 Oe (1750 A m-r) the wide domains have vanished (fig. 8). Within the patches one system of narrow laminae increases in width whereas another system shrinks in width. The narrow laminae vanish at about 62 Oe (4900 A m-r). By an annealing treatment the domain patches as well as the number of wide domains is reduced considerably. In some regions only two domains are observed over the whole width of the ribbon.
Fig. 6. Domain structure with vanishing narrow laminae for an applied field of 1750 A m-l (Metglas).
2.3.3. Co70FepYiPB~6 (Vakuumschmelze Hanau) This amorphous Co-alloy is characterized by a vanishing magnetostriction constant. Characteristic domain patterns of the as-quenched specimen are
Fig. 7. Domain structure of an as-quenched Fe 80 Bzo-ribbon showing wide and narrow laminae at zero-applied field.
58
Fig. 8. Narrow H=4900Am-‘.
H. Kronmiiller et al. /Magnetic
and wide domains
of Fego&o
under
properties of amorphous ferromagnetic
the influence
of an applied
alloys
field; (a) H = 0 A m-l;
(b) H = 1750 A m-l;
(c)
H. Kronmtiller et al. /Magnetic
Fig. 9. Domain
structure
of an asquenched
properties of amorphous ferromagnetic
Co 70FegSigB16-ribbon
shown in fig. 9. Together with wide laminae oriented perpendicular to the ribbon axis we observe a stripe domain structure which appears to be superimposed in the wide domains. After an annealing treatment at 500 K the stripe domains have vanished and the wide
Fig. 10. Domain
structure
of a Co7OFegSiqB16-ribbon
showing
wide laminae
alloys
59
and weak stripe domains.
domains show a marked tendency to become wider and to align parallel to the ribbon axis (see fig. 10). Under an applied magnetic field oriented perpendicular to the stripe domains these vanish at about 8 Oe (650 A m-l). With a magnetic field applied
after annealing
for 27 h at 500-515
K.
60
H.
Kronmiilleret al. /Magneticpropertiesof amorphousferromagnericalloys
parallel to the stripe domains the wide laminae vanished at about 40 Oe (3200 A m-l) whereas the stripe domains still were present at 88 Oe (7000 A m-l).
3. Interpretation
relation lutl ut - loci u, = 0 .
(4)
By means of eq. (4) we now may derive a number of important properties of the magnetization curve and the domain structure.
of domain structures 3.2. Magnetoelastic coupling energy
3.1. Directions of easy magnetization The arrangement of domains in the as-quenched alloy is characterized by a composition of wide laminae and patches of narrow laminae. The fact that the narrow laminae react mainly on magnetic fields applied perpendicular to them shows that this domain pattern corresponds to a closure domain structure. A model of domain structures derived from these sug gestions is given in fig. 11. According to these results we assume that within the wide laminae the easy directions lie within the plane of the ribbon whereas within the patches the easy directions are oriented perpendicular to the ribbon. As discussed previously the anisotropy energy responsible for these easy directions is suggested to be the magnetoelastic coupling energy. According to eq. (3) in the case of a positive magnetostriction, tensile stresses produce an easy axis parallel to the tensile axis whereas compressive stresses lead to an easy axis perpendicular to the compression axis. From an elastomechanical point of view the stress axes should lie parallel to the ribbon. Therefore wide laminae are attributed to regions of tensile stresses, ut , and narrow laminae patches to regions of compressive stresses, uC. As shown by Albenga [ 181, the volume averages of internal stresses vanish. By introducing the partial volume fractions ut and uC of tensile and compressive stresses this leads to the following
For a random distribution of domains (($) = 3) the volume average of the magnetoelastic coupling energy is given by K, = $Qlatlut
+ IU&,) = &W,i%.
(5)
If no other anisotropy energies are present K, is directly measured by the magnetization energy. 3.3. Magnetization processes Due to the rather inhomogeneous domain structure of the as-quenched alloys the magnetization is controlled by at least three main types of magnetization processes. (i) In the low field region (zero internal field) magnetization takes place by 180”-dw displacements of wide domains. Accordingly the initial magnetization occurs in the regions of tensile stresses. (ii) The narrow laminae patches in a first stage are magnetized by a growing of the energetically favourable closure domains as indicated in fig. 6. A full magnetization of the closure domains is obtained for the magnetic field HI = Kc/J,,
(6)
where K, = ~Aslocl
(7)
corresponds to the anisotropy energy within the closure domains. (iii) In a second stage within the narrow laminae patches rotational processes take place as shown in fig. 12. It may happen that stage I and stage II take place simultaneously if the internal magnetic field increases at external fields smaller than HI. The spontaneous magnetization in the narrow laminae is aligned parallel to the applied magnetic field for Fig. 11. Schematic domain model showing wide laminae narrow domains with their closure structure.
and
HII = 2K,/J,.
(8)
H. Kronmdller
et al. /Magnetic
properties of amorphous ferromagnetic
H
-
Hr
b
Fig. 12. Magnetization process within narrow domains. (a) Growing of the closure domains at small fields (H < HI). (b) Rotation of the spontaneous magnetization within the narrow laminae for HI < H < HII.
3.4. Domain wall width of narrow domains 3.4.1. Ni40Fe40P14B20 and FegoBzo alloys The domain wall width of the stray-field free laminar domain structure of the Landau-Lifshitz is given by DO = (2TO~BfKc)“2.
type (9)
Here Tn = 4(AK,)l12 corresponds to the specific wall energy in the compressive regions and To denotes the thickness of the ribbon. The exchange constant, A, may be determined from the spinwave stiffness constant, D,, , as discussed in section 6. From (9) we obtain for the magnetic anisotropy energy, K,, within the closure domains and for Yn K, = 64AT;/D;,
(10)
TB = 32ATO/D&
(11)
The numerical
61
(lo), and (11) are given in table 1 for Ni4eFe4ePr4B6 and FeseB2e. From our results for K, we may determine the amplitude of the compressive stresses from eq. (7). With h, = 11 X 10e6 [ 191 we find for Ni40Pe4ePr4Be Us = 0.17 X 10’ N m-’ and for FeseB2e with h, = 30 X lop6 [22] we obtain u, = 1.1 X lo8 N rnp2. It should be noted that the internal stresses are found to correspond to -1% of the shear modulus.
a -
alloys
3.4.2. Co 70Fe5SigB16-a110ys Alloys of this composition are characterized by a vanishing magnetostriction. The domain structures in these cases are determined by shape and structural anisotropies. According to fig. 9 the domain structure is composed of wide domains and so-called stripe domains [20-221 with a wavelength of h - 6 pm. The appearance of a stripe domain configuration points to the fact that we are dealing with an oblique or perpendicular anisotropy which may be due to a structural anisotropy. As we may assume that the shape anisotropy is much larger than the oblique anisotropy, the spontaneous magnetization is tilted out of plane by only a small angle Oe. This leads to the configuration of wide domains shown in fig. 13. In the case of a homogeneous magnetization within the wide domains, surface charges would produce a large magnetostatic energy. In previous papers it has been shown [2 1,231 that these surface charges are suppressed by the formation of a stray-field free stripe domain structure. The conditions for the formation of such stripe domains in the case of a perpendicular anisotropy energy, KI, are the following:
values derived for K, and Yn from (7)
Table 1 Material parameters of the domain structure and of the domain walls in Metglas and FegoBZo. into SI-units the following relations hold: 1 Oe = (103/4n) A m-t, 1 G = lop4 T, 1 erg cm-l J m-‘, 1 erg cmm3 = l/10 J mv3
For a conversion = lo-’ J m-l;
of the cgs-units 1 erg cmP2 = 10e3
HI
47rM,
(Oe)
(G)
To (pm)
Do (pm)
A (erg cm-‘)
/(HI) KC‘(Do) (erg cmP3)
6B (crm)
YB (erg cme2)
Ni4oFe4oP14B2o
20
8000
20
4-6
3.1 x 10-7
0.62 X lo4 2.8 x 103
0.314
0.11
FesoB20
90
1600
20
3
5
6.0 4.9
0.10
0.64
x 10-7
X lo4 x 104
H. Kronmiiller et al. /Magnetic
62
properties of amorphous ferromagnetic
susceptibilities x0, large Rayleigh constants CY,and small coercive fields Hc. Fig. 14 presents the magnetization curve in the low field region and in fig. 15 the dependence of the quantity M/H on H is shown. From the linear dependence of M/H on H follows the validity of Rayleigh’s law. Previously the parameters x0, (Y,HG and H, were determined on the assumption that the displacement of the dw may be described by potential theory [24-271. Within the framework of this model it is assumed that the total force acting on a dw is given
Fig. 13. Model of wide domains and superimposed stripe domains in the case of a perpendicular easy direction.
To > 2n(A/K1)“’
by
where To corresponds to the thickness of the ribbon. In the case where K&p&) << 1 the wavelength of the stripe domain is given by XsD = 4n(A/K#‘*.
(12)
As in the case of an oblique anisotropy the only important term is K. sin* Oo which is connected with the component of MS perpendicular to the ribbon; we obtain X,,
= (4n/sin O,)(A/Ko)‘/*
(13)
.
The anisotropy constant K,, may be determined from the critical field for which the stripe domains and the wide domains vanish if applied perpendicular to the laminae. With H,, = 5 Oe we obtain from H,, = 2K,,/J, the anisotropy constant K. - 2.5 X lo3 erg cmA3 . With a reasonable value of A = 10m6 erg cm-’ we have to assume 00 = 20” in order to explain the observed wavelength of -6 pm.
4. The characteristic
alloys
parameters of the Rayleigh
P(Z)
= C
p(Z
-
Zi),
(19
i
where the sum extends over all individual pinning forces p(z - Zi) resulting from pinning centers at position Zi. The coordinate axis z is taken to be perpendicular to the dw. In general eq. (15) will lead to a fluctuating field of force and it may be shown [24,25] that the probability f(p) for the existence of a force between p and p t dp is given by the so-called normal distribution (see fig. 16) f(p) = (2nB)-1’2 exp(-p2/2@, where the correlation D/2
s
(16)
function B is defined as
cm}*dz.
(17)
-D/2
(Fn = dw area moving coherently, D = distance between neighbouring dws). By means of (16) the characteristic statistical properties of the field of force may be determined. Such properties are the
region
At small magnetic fields the magnetization obeys Rayleigh’s law M(H) = x,,H + d-f*
curve
(14)
where x0 denotes the initial susceptibility and OLthe Rayleigh constant. The magnetic field, HG , below which Rayleigh’s law holds in general is of the order of magnitude HG 5 OSH, where H, corresponds to the coercive field. Soft magnetic materials are characterized
1
18'
LTI 25 20 I5 -
0123456799
by large
Fig. 14. Magnetization (Fe40Ni40P14B6-ribbons)
10
curve of asquenched in the Rayleigh
&,H
Metglas region.
[IO-‘Tl
H. Kronmiiller et al. /Magnetic properties of amorphous ferromagnetic alloys
1 0
12
3
Fig. 15. The quantity the Rayleigh region.
4
5
J/p@
6
7
6
9
10
11
of Metglas as a function
poH CIO-‘TI
of H in
average value of the maxima of the force, the average wavelength, he, of P(z) and the average value, l/(dP/dz), of the reciprocal slope of P(z) at the dw positions where P(z) = 0 holds. Previously it has been shown that the potential theory leads to the following relations between x0, H, and QI(in Gaussian units):
xo& = Msvt -t x;/(olM,)
2
(ln
D/ho)l'* ,
= u,(37rll6)(ho/2D),
cuH,/xo = (g/3&)
(19)
[ln(D/~e)] ‘I2
(20)
In (18)-( 19) ho corresponds to the average wavelength of the field of force. From statistics it follows that ho - 46 n . All other quantities in the above relations x0, H,, CY,the partial volume vt of the tensile stresses and the laminae width D are known from
irrev. P(z)
Fig. 16. Definition of characteristic force acting on a domain wall.
parameters
of the field of
63
experiment. The partial volume vt appears in (18) and (19) because in the Rayleigh region dw movements take place only within regions of tensile stresses. In table 2 we give the experimental parameters of NiFeBalloys required for a test of eqs. (18)-(20), and fig. 17 shows OlH, as a function of x0. Because of the small variation of ln(D/ho) this function according to (20) should correspond to a straight line with a slope (g/3&) [ln(DlMl ‘I2 . The dashed line in fig. 17 corresponds to the slope obtained for D and 6n as determined for Fe40N&0P14B6 (see table 1). The agreement between experiment and theory as demonstrated by fig. 17 is surprisingly good for a statistical theory. The experimental points obtained for the CoFeSiB-alloys deviate considerably from the theoretical prediction thus indicating that in magnetostrictionless materials other, probably surface effects, are dominant for the magnetization processes.
5. Law of approach to ferromagnetic
saturation
5.1. General remarks Whereas domain patterns are suitable for a study of long range stresses with wavelengths h > 5 pm short range stresses with wavelengths 10 A < h < 1 pm may be studied by means of the field dependence of the high-field susceptibility. The theoretical background of this method has been developed in numerous papers [28-331, [S-7]. In amorphous alloys the spin moments are not aligned parallel to the internal magnetic field because of the following three perturbations: (i) as a consequence of the random distribution of atoms in ideal amorphous alloys with Bernal structure, intrinsic material properties, such as the local spin-orbit and exchange couplings as well as the modulus of the spontaneous magnetization, IM(r)l fluctuate randomly. Due to these intrinsic fluctuations the orientation of the spontaneous magnetization M(r) deviates in general from the direction of the internal magnetic field. Previous theoretical investigations [7], however, have shown that the effect of these intrinsic fluctuations on the law of approach to saturation is negligible in the case of transition metal-metalloid alloys. The same result holds for the influence of non-magnetic precipitates [34]. (ii) In real amorphous alloys there exist structural
H, Kronmiiller et al. /Magnetic
64
properties of amorphous ferromagnetic
alloys
Table 2 Characteristic parameters of the Rayleigh region of amorphous alloys. The material parameters are given in “Gaussian” units in the upper rows and in SI units in the lower rows. In order to convert the material parameters from “Gaussian” units into SI units, Js has to be multiplied by 4n X 10e4, H, by no X 10e4, x0 by 4n, and oi by 4n X lo4 0.Q = 1.256 x lo6 (V s A-’ m-l) (kc) Js (I’)
Hc n@c
(‘I)
/ /
t
(G 0eC2)
O1(T-l)
l-l
4.8 x 10’ 6.03 x lo2
3.16 X lo3 3.97 x 108
1.273 16.00 x 10-l
1.32 x 10-l 1.32 x 1O-5
9 x 10’ 1.13 x 103
2 x 103 2.51 X lo8
1.034 13.00x
1.08 x 10-l 1.08 x 1O-5
9.3 x 101 1.17 x 103
2.85 X lo3 3.58 X lo8
0.915 11.50 x 10-r
8.6 8.6
x 10-2 x 10-e
1.09 x 102 1.37 x 103
2.5 X lo3 3.14 x 108
0.796 10.00 x 10-l
6.5 6.5
x 1O-2 x lO-‘j
1.13 x 102 1.42 x lo3
2.74 X lo3 3.44 x 108
0.6685 8.40 x 10-l
4.25 x 1O-2 4.25 x 1O-6
1.2 x 102 1.51 x 103
6.58 X lo3 8.27 x lo8
0.43 5.40 x 10-r
4.9s x 10-Z 4.95 x 10-e
1.3 x 102 1.63 x lo3
7.69 x lo3 9.66 x lo8
0.533 6.70 X 10-l
1.98 x 1O-2 1.98 x lo@
2.5 x lo2 3.14 x 103
1.6 x lo4 2.01 x 109
0.482 6.06 x 10-r
4.29 x 10-Z 4.29 x 10-e
4.3 x 102 5.40 x 103
2.6 x lo3 3.27 x lo8
0.433 5.44 x 10-l
9.24 x 10-Z 9.24 x 10-e
1.1 x 102 1.38 x lo3
4.1 x 103 5.15 x 108
10-r
deviations from the ideal Bernal structure giving rise to internal elastic stresses. Due to the magnetoelastic coupling energy these internal stress fields produce
200
xo
x 1O-2 x lo@
0.621 7.8 x 10-l
4.2 4.2
(G Oe-‘)
(Oe)
X F&O NiJo BZO
x FeroNlra 8x1
an inhomogeneous spin arrangement as shown qualitatively in fig. 18 for a dislocation dipole. (iii) In addition to static inhomogeneities at finite temperatures, T,dynamic spin inhomogeneities exist due to thermally excited spinwaves. Under the action of an applied field, H, the deviating spins rotate into the direction of the applied field. This rotation process is field dependent and may be described by the empirical law of approach to saturation:
J(T, H> = Js [ 1 - (T/To>~'~ 1 + AJpara alI2
-H’12-g-$’
Fig. 17. Test of the self-consistency relation (20) of the potential model for dw displacement in amorphous alloys Fego.-JVi,B2o and CoFeSiB-alloys (compare table 2).
al
a2
(T H)
(21)
where J, corresponds to the absolute saturation polarization J, = ~~~~ = J(0, -), To denotes a characteristic temperature, and the parameters ap describe the effects of the microstructure. The different terms
H. Kronmiiller et al. /Magnetic properties of amorphous ferromagnetic alloys
-
_\---,‘-+
--
-‘-_;
_
-
A---
field range we plot J(T, H) as a function of l/W. Figs. 19 and 20 show the results of such plots for spin-cooled amorphous alloys. In the case of a Fe40Ni4ePr4B6 alloy the approach to saturation is described by a H-’ -term for H < 2000 Oe. Somewhat more complicated is the behaviour of a Fe~eNLreB2e-alloy which in the field range 100 < H < 300 Oe obeys a l/H-law and in the field range 300 2000 Oe the J(T, H) versus l/H”-diagrams show a steep increase (see figs. 19 and 20) due to the spinwave term of (21). For an analysis of this region we have eliminated the spinwave contribution to J(T, H) by extrapolation of the microstructural effects to higher field. The para effect is then given by
-
-
-_
65
-
I+
Fig. 18. Schematic representation of the spin arrangement in the neighbourhood of a dislocation dipole.
= J(T, HI -
AJp,
Js(T, 0) + q 3
where JS(T, 0) corresponds
2
(23)
to the spontaneous
polari-
in (2 1) have the following meaning:
(1) the first two terms correspond to the effect of thermally excited spinwaves and to the so-called paraprocess AJ,,,(T, H) [35]. (2) Point defects with stress fields varying as u a l/r3 give rise to the l/H’12-term. (3) Straight dislocation dipoles with stress fields u a l/r2 lead to a l/H-term if the distance Ddtp between the two dipole dislocations is smaller than the exchange length: Hdip <
Kfil
= (2~4/~cf~)“~e
(22)
(4) The l/H2 -term may have different origins. (i) Single dislocations with stress fields varying as u a l/r or dislocation dipoles with Ddip > Kfil. (ii) Magnetic anisotropy due to long range stresses with wavelength h >pm. 5.2. Experimental
results
ECq.(21) shows that a determination ofJ,(T, 0) = Js [ 1 - (7’/re)3/2 ] for zero field in general is rather diffvult because in the high field region the para becomes important whereas at lower effect AJ,, fields the microstructural effects of the type H-” are prevailing. Therefore different methods are used in order to analyse these two field ranges. In order to determine the microstructural terms in the lower
1
0
C
20
LO
60
f&H
CT-“21
Fig. 19. The magnetization curves of Fe40Ni40Pt4B6 for different temperatures as a function of l/&H) [ 361.
H. Kronmtiller et al. /Magnetic
66
properties of amorphous ferromagnetic
alloys
JCTI
1.20 16 t I.16
0.4 1.12 I
0
200
100
I
300
Fig. 20. The magnetization curves of Fe46Ni4uB2u ent temperatures as a function of l/(r&12.
Ip_Fil'[T'l
for differ-
zation obtained by the extrapolation of l/H” --f 0, and the third term corresponds to the correction of the microstructural effects (compare insert in fig. 19). According to Holstein-Primakoff [35] the para effect may be written as J para = c(n
(24)
fHP
with the Holstein-Primakoff
0123456789
I
4oo
IO
f"p [T”21
Fig. 22. The para effect hlpara = n&Mpara for Fe4uNi4uB20 represented as a function of the Holstein-Primakoff function fHp for different temperatures.
cases the para effect is found to depend linearly on fHp. From the slopes of the straight lines we obtained the functions C(7) as represented in fig. 23. For temperatures T < 200 K a linear temperature dependence was found and the coefficients c = dC/dT are given by c = 1.3 X lo-’ for Fe40NLroP14B6 and c = 0.89 X 10m5 for Fe40Ni40B20 in units of T112 K-‘.
function 5.3. Interpretation of microstructural effects
and the temperature function C(T) proportional to T. In fig. 21 and fig. 22 the para effect AJ,,, is represented as a function of fHp for both alloys. In both
.a2
As discussed in section 5.1 internal stresses varying according to l/r give rise to characteristic field dependences in powers of l/H. The experimentally determined exponent of the 1/E-I”I-law therefore indicates which kind of internal stresses exist in amorphous alloys. Previous investigations of the effect of internal
295 K .10-3
5-
LFe40 Wo PM 6%
..‘-..-.x-x OY’IQ 0
2.0
L.0
91 K
/d /
)
Ix-
u I 6.0
1 8.0
L.2K I fH1?[T"*l 1
.3g. 21. The para effect Upara = n&Vfpara for Fe4ONi40Pt4Be represented as a function of the Holstein-Primakoff function fHp for different temperatures.
0
I
A0
1
80
Fig. 23. The temperature and Fe40Ni40BZO.
I
120
function
I
160
I
200
1
240
I
T CKI
C(T) for Fe4uNi4uP14Be
H. Kronmtiller et al. /Magnetic properties of amorphous ferromagnetic alloys
stresses on the law of approach to saturation were based on the concept of lattice dislocations with their strength characterized by a discrete Burgers-vector b and a well-defined line direction. Obviously this concept cannot be applied to amorphous materials without any modification. Here it is referred to the work of Krijner [37 ] who has shown that any internal stress state can be represented by quasi dislocations, i.e., in this generalized continuum theory the concept of discrete Burgers vectors as in crystal lattices is replaced by the concept of a dislocation density where the effective strength, b,rf, of a quasi-dislocation may correspond to a “Burgers vector” much less than the nearest neighbour atomic distances. Quasi-dislocations in amorphous materials may result from mass density fluctuations which were quenched-in during the rapid cooling process. A simple two-dimensional model of such a mass density defect producing dislocation stresses is shown in fig. 24. Here it was assumed that vacancy-like defects have agglomerated during the quenching process thus forming a diluted zone which leads to a local collapse of the lattice in these regions. This lattice contraction induces elastic deformations in the surrounding amorphous structure. Another type of stress sources may result from planar precipitates of metalloid atoms which induce compressive stresses. This type of structural inhomogeneities corresponds to the counterpart of the vacancy type precipitations. Both types of defects may be considered as dipoles of quasi-dislocations with a stress field varying as l/r’. In section 5.1
it was discussed that stress fields of this kind in fact give rise to a l/Z&term in the law of approach to saturation if the stress centers forming the dipoles have a distance, Ddip, smaller than the exchange length KE’ . The general relation for the AZ-effect due to quasidislocation dipoles of density Ndtp is given by [41]
(26) where F= 1.67 denotes a numerical factor which takes into account the statistical distribution of orientations of quasi-dipoles. The elastic properties of the amorphous material are characterized by the shear modulus G and Poisson’s constant V. In the case of Metglas Fe40Ni40Pr4B6 we have G = 4.93 X 101’ erg cmm3 E 4.93 X 10” Jme3, and v = 0.37, ref. [38]. In (26) F. takes care of the effect of one isolated quasi-dislocation dipole, and is given by FO = M&KHDdip)
’ $(KHDdip)
+Ko(K~Ddip) K1 (KHDdip)
+ 70 -
0.51,
(27)
where y. = 0.5772 corresponds to Euler’s constant, denotes the exchange length given by (22), and K. and K1 denote the modified Bessel functions of the second kind and of order 0 and 1. The function Fi:,nt takes into account the interaction between neighbouring dipoles. If RiZ corresponds to the distance between the ith and jth dipole and the condition Rij > Ddip holds Fint may be written as KH
Fint = C i
eij(KO(KHRij)[ZO(KHDdip)
KHDdip
--Il(KHDdip)
2
+F
Fig. 24. Schematic model for the formation of quasidislocation dipoles in amorphous alloys by the agglomeration of vacancy-type point defects in planar regions.
61
-
11
Kl(KHRij)[Io(KHDdip)
- 11)
(28)
where eij = 1 for dipoles with the same polarity, and Eij = -1 for dipoles of opposite polarity. IO and Zr denote the modified Bessel functions of the first kind. The functions F. and Fint were calculated as a function Of the parameters KHDdip and KHRii for a square lattice of lattice constant DdiD, i.e., there are
H. Kronmiiller et al. /Magnetic properties of amorphousferromagnetic alloys
68
four nearest neighbours at a distance (Rii)r = Ddip, four next nearest neighbours at (Rii)z = d2 Ddip, etc. We furthermore assume that there is a strong correlation in orientation and polarity of all dipoles up to the nth shell of the dipole lattice. Under the assumption that all dipoles are due to an agglomeration of vacancytype defects we have to choose in (28) eij = 1. In fig. 25 the functions Fo + Fint and Fo are represented for n = 11 which corresponds to no = 60 neighbours. The function FO + Fint may be described by the following three regions: (i) region I: KHDdip < 0.12 m
FO +Fint
1
AJ=----
-
$KhDitp(l
9 FG’hzNdipb&
477 64
(1 - v)~A
(ii) region II:
AJ=-
X
region
111:
1.22
0.12
D&p(l +“o)$
(29)
Fig. 25. The functions ln(K@dip), calculated with no = 60 dipoles.
1.95
20.1
X+, h,,p
FO and FO + Fint in dependence for a square lattice
of dipole
of clusters
1.22 < KHDdip < 5
9 FG2X2N s dip b= eff 1.6 _l_ 4 (1 - V)%l!fS H2 ’
(iii)
AJ=-
+ no),
(30)
KHDdip > 20.
9 FG~~=N. s drp b2eff i 4 (1 - v)=& 9
{hl(iKHDdip)+ 70
-
OS}
.
(iii) At very large magnetic fields, in region III, between dipoles becomes negligible and AJ is due to isolated dipoles. For larger values of (Rtj)l and smaller values of 12this result holds for smaller vdues of KHDdip . For a test of the above results we have represented in fig. 26 the experimental quantity AJH’ which should lead to a function linear in H for small values of H and a constant for larger values of H. Just this prediction in fact is observed for FedoNiQoBzo. A quantitative calculation leads to the following parameters (using the same values for G and v as for Metglas): KHDdip > 20, the interaction
(31)
It should be noted that the field range of validity of the l/H-power law increases with decreasing numbers n of shells considered in Fint . The range of validity is large for (Rtj)l - Ddip and (Rij)l + 00, but is reduced for all other values of (Rtj)l . On the other hand the results for regions II and III remain nearly unchanged if we choose n > 11 whereas in region I AJ is proportional to the number of dipoles no within the dipole cluster, The main conclusion to be drawn from these results are (i) in region I we obtain the well-known l/Hdependence of dislocation dipoles (see insert in fig. 25). (ii) In region II the magnetic interaction term leads to an enhanced approach to the l/Hz -dependence. The transition from a l/H- to a l/H2-law takes place in the interval 0.12 < KHDdip < 1.22.
Ddip = 245 A, Ndip = l/Dzip = 1.8 X 10” b eff = 1.9 A,
and
cm -2 ,
n = 11.
In the case of Fe40Ni40P14B6 within the whole field range /..L,,H < 0.2 T a linear increase of AJH2 is observed. This result is in agreement with measurements of Kazama and Kameda [39]. In order to increase the field range within which the l/H-law holds we assume that in Metglas a correlation only exists between nearest neighbour dipoles (n = 1). In this case the l/H-law is valid up to KHDdip = 0.6, and we obtain as an upper limit of the dipole width Ddlp < 0.6 KE1 @OH = 0.2 T) = 35.4 a. This corre-
69
H. Kronrmiller et al. /Magnetic properties of amorphous ferromagnetic alloys
1251.0
0.9 1.15 -
0
0
0.01
0.02
0.03
0.04
0.05
0.06
Fig. 26. The quantity aJH2 as a function of rfl
0117#fCTl for
1CQO
2000
3000
4000
Fig. 27. The temperature dependence of the spontaneous polarization in Fe40N&P14B6 and Fe40NiaoB20 alloys.
Fe4oNkoB2o.
sponds to a dipole density Ndip = l/Diin = 8 X 10” -‘. With b eff = 2.5 A we then are able to reprolice the experimentally observed values of AJ. Obviously in the Fe40Ni40P14B,s-alloy investigated Ddi,, and the correlation effects are smaller than in Fe40Ni40B20 : probably because of different quenching rates during the quenching procedure. In fact the observed domain structures support this assumption because in Fe40Ni40B20 we have found nearly no stress-induced domain pattern. Furthermore we observe in annealed Fe40Ni40Pr4B6 a l/H2 term, too, which is attributed to increased dipole widths and a larger correlation between neighbouring dipoles. According to these results a self-consistent interpretation of the law of approach to saturation seems to be possible on the basis of stress sources which exert internal stresses similar to those of straight dislocation dipoles. The fact that the dislocation strength b,ff is found to be of the order of magnitude of the interatomic distances strongly supports this interpretation. Finally it should be noted that our experimental result of the transitions from a l/H to a l/H2 law is supported by recent investigations of the magnetic small-angle neutron scattering in Fe40Ni40PloCio-alloys by Gijltz [40]. 5.4. Temperature dependence of the spontaneous polarization A comparison of the temperature dependence of J, with the theory of thermal spinwaves requires a careful determination of J, for zero applied field. As outlined in section 5.2 the spontaneous magnetiza-
tion J,(T, 0) for H = 0 can be determined by an extrapolation of the law of approach to saturation from the lower field range into the high field range. According to figs. 19-20, we obtain J, (T, 0) from the intersections of the l/H”-plots ofJ(T, H) with the coordinate axis at l/H” + 0. By this method we eliminate the microstructural term as well as the para effect of the spinwaves. The results of such an extrapolation are shown in fig. 27 where J,(T, 0) is plotted versus T312. In both alloys a T 3/2-law was found to be valid up to one half of the Curie temperature. Taking into account our results for the spinwave para effect we obtain for the field and temperature dependence of J,:
J&T H>= Js 11- (T/To)~‘~1 + cTfHp,
(32)
where fHp corresponds to the Holstein-Primakoff function [35]. The experimental parameters To and c are summarized in table 3. As is well known Bloch’s T3/2-law holds if the spinwave energy spectrum, ek, obeys a square dependence ek = D,,k'
,
(33)
where D,, corresponds to the so-called spinwave stiffness constant and k denotes the spinwave wavenumber. The validity of Bloch’s T312-law up to high temperatures can only be understood if the k4 terms in ek play a minor role due to a high density of states for the low energy spinwaves. The spinwave stiffness constant D,, is related to the parameter TO of Bloch’s law by
Ds,
=-& [Z~~;;B]2’3kB~o,
H. Kronmiiller et al. /Magnetic
70 Table 3 Material constants
of amorphous To (K)
F’QoNi4oPldb FedGoB
FegoBZo
]161
properties of amorphous ferromagnetic
alloys D SP (meV A2)
840
90
1300
191
1270
91
and the exchange constant A is given by A =Ms(%
~)&&‘~B
with g corresponding to the g-factor. Our results for To, D,, and A are given in table 3. References [l] H.-R. Hilzinger, A. Mager and H. Warlimont,
[2]
[3] [4] [S]
[6] [7] [8] [9]
[lo] [ 1 l] [ 121 [13] [ 141
[ 151 [16]
alloys
J. Magn. Magn. Mat. 9 (1978) 191; (Proc. Arbeitsgemeinschaft Magnetismus, Freudenstadt and Bad Nauheim 1978). W. Wolf, J. Magn. Magn. Mat. 9 (1977) 200; (Proc. Arbeitsgemeinschaft Magnetismus, Freudenstadt and Bad Nauheim 1978). F.E. Luborsky, PG. Frischmann and L.A. Johnson, J. Magn. Magn. Mat. 8 (1978) 318. C.D. Graham and T. Egami, Ann. Rev. Mat. Sci. 8 (1978) 423. R.G. Henderson and A.M. de Graaf, in: Amorphous Magnetism, eds. H.D. Hooper and A.M. de Graaf (Plenum Press, New York-London, 1973) p. 331. H. Kronmiiller and J. Ulner, J. Magn. Magn. Mat. 6 (1977) 52. M. FHhnle and H. Kronmiiller, J. Magn. Magn. Mat. 8 (1978) 149. J.J. Becker, IEEE Trans. Magn. Mat. 11 (1975) 1326. Y. Obi, H. Jujimori and H. Saito, Japan. J. Appl. Phys. 15 (1976) 611. M. Takahashi, T. Suzuki and T. Miyazaki, Japan. J. Appl. Phys. 16 (1977) 521. S. Tsukahara, T. Satoh and T. Tushima, IEEE Trans. Mag. 14 (1978) 1022. G. Dietz, J. Magn. Magn. Mat. 6 (1977) 47. H. Kronmiiller, R. Schifer and G. Schriider, J. Magn. Magn. Mat. 6 (1977) 61. A. Hubert, J. Magn. Magn. Mat. 6 (1977) 38. B. Griiger and H. Kronmiiller, J. Magn. Magn. Mat. 9 (1978) 203. G. Schroeder, R. Schifer and H. Kronmiiller, Phys. Stat. Sol. (a)50 (1978).
A (J m-‘)
c
(T=OK)
@/K)
3.1
1.3
x 10-12
x 10-S
8.07 x lo-l2
0.89 X 1O-5
5
_
x 10-12
[ 171 J. Kranz and A. Hubert, Z. Angew. Phys. 15 (1963) 220. [ 181 G. Albenga, Atti Accad. Sci. Torino, Classe Sci. Fis., Mat. Nat. 54 (1918/19) 864. [19] T. Egami and P.J. Flanders, AIP Conf. Proc. 29 (1976) 220. [20] Y. Murayama, J. Phys. Sot. Japan 21 (1966) 2253. [21] A. Holz and H. Kronmiiller, Phys. Stat. Sol. 31 (1969) 787. [22] M.W. Muller, Phys. Rev. 122 (1961) 1485. [23] H. Kronmiiller, Z. Angew. Phys. 32 (1971) 49. [24] H. Trguble, in: Moderne Probleme der Metallphysik, vol. 2, ed. A. Seeger (Springer-Verlag, Berlin, 1966). [25] K.-H. Pfeffer, Phys. Stat. Sol. 21 (1967) 857. [26] H. Kronmiiller, Z. Angew. Phys. 30 (1970) 9. [27] [28] [29] [30] [31] [32] [33]
H.-R. Hilzinger and H. Kronmliller, J. Magn. Magn. Mat. 2 (1976) 11. W.F. Brown, Jr., Phys. Rev. 58 (1940) 736; 60 (1941) 132. A. Seeger and H. Kronmiiller, J. Phys. Chem. Solids 12 (1960) 298. H. Kronmiiller and A. Seeger, J. Phys. Chem. Solids 18 (1961) 93. H. Kronmiiller, Canad. J. Phys. 45 (1967) 631. H. Kronmiiller, Int. J. Non-Destructive Testing 3 (1972) 315. H. Kronmiiller, J. Appl. Phys. 38 (1967) 1314.
[34] M. Fahnle, J. Magn. Magn. Mat. 8 (1978) 2.57. [35] T. Holstein and H. Primakoff, Phys. Rev. 58 (1940) 1098. [36] H. Kronmiiller and H. Grimm, J. Magn. Magn. Mat. 6 (1977) 57. [ 371 E. KrGner, Kontinuumstheorie der Versetzungen und Eigenspannungen (Springer-Verlag, Berlin-GiittingenHeidelberg-New York, 1958). [38] P.H. Chang, M. Grimsditch, A.P. Malozemoff, W. Senn and G. Winterling, Solid State Commun. 27 (1978) 617. [39] N. Kazama and M. Kameda, AIP Conf. Proc. (USA) no. 34 (1976) 307. [40] G. Gijltz, private communication. [41] H. Kronmiiller, to be published in a special issue: Advances in Magnetics, IEEE Trans. on Magn. honouring W.F. Brown Jr.
MAGNETIC PROPERTIES H. KRONMULLER,
OF AMORPHOUS FERROMAGNETIC
ALLOYS
M. FAHNLE, M. DOMANN, H. GRIMM, R. GRIMM, B. GROCER
Institut fiir Physik am Max-Planck-Institut fiir Metallforschung,
Stuttgart, Fed. Rep. Germany
Received 11 April 1979
For the application of amorphous, ferromagnetic alloys as soft magnetic materials a detailed knowledge of the domain and thermodynamic properties is required. A review of the present state of the research work will be given. It is shown that the displacement of domain walls is described fairly well by the conventional potential theory developed previously for crystalline materials. Within the framework of the theory of micromagnetism the effect of quenching stresses on the domain structure as well as of short range stresses on the magnetic microstructure are discussed in detail,
structure, magnetization processes, the magnetic microstructure
1. Introduction
talline materials. An important feature of amorphous alloys is, furthermore, the absence of any grain or phase boundaries which are known to act as strong pinning centres for dws in crystalline materials. In amorphous alloys therefore one of the main sources of pinning centres is eliminated. Nevertheless the initial permeability of amorphous alloys in general is a factor of 5 smaller than that of crystalline materials. However, for finite magnetic fields of H w 20 mOe (0.02 A cm-‘) the permeability is comparable with that of crystalline materials [l-3]. On the other hand the coercive fields are in general found to be smaller than those of crystalline soft magnetic materials. The origin of this rather peculiar behaviour must be attributed to properties of the domain structure as well as of the microstructure. Extensive investigations have been performed with respect to intrinsic properties of amorphous alloys; e.g., the spontaneous magnetization and the Curie temperature have been investigated for many alloys (see review articles by Egami and Graham [4], Luborsky et al. [3], and Hilzinger et al. [I]). Only a few investigations, however, are known which deal with an interpretation of the characteristic properties of the hysteresis loop. For a solution of these problems investigations in the following fields are required: (1) domain patterns; (2) magnetization processes; (3) magnetic anisotropy energy;
Amorphous, ferromagnetic alloys have turned out to be a promising new soft magnetic material with attractive mechanical and magnetic material properties. Some of the outstanding properties are the following: (1) small coercive fields; (2) large permeability; (3) small magnetic losses; (4) small magnetic anisotropy; (5) large critical shear stress. As is well known in crystalline materials lattice imperfections act as rather effective sources for the destruction of excellent soft magnetic properties, This undesired effect is in general due to strong magnetoelastic interactions between domain walls (dws) and the internal stresses of lattice defects. Therefore even small plastic deformations of crystalline magnetic materials lower the permeability considerably. Generally a careful annealing treatment is required in order to reduce these pinning effects of microstructural defects. In addition the magnetostriction is reduced to approximately zero by developing appropriate alloys. Well known examples of these alloys are Deltamax, Permalloy and Supermalloy. As plastic deformations are rather destructive for soft magnetic materials amorphous alloys with their large critical shear stresses have significant advantages in comparison with crys53
54
H. Kronmiiller et al. /Magnetic
properties of amorphous ferromagnetic
(4) interaction between microstructure and domain walls; (5) characterization of pinning centres in amorphous alloys. The aim of this paper is a concise review of the above mentioned topics. For a study of domain structures, macroscopic magnetization processes and anisotropic energies we have applied the method of the magneto-optical Kerr effect, and for a study of the microstructure we have investigated the Rayleigh region and analyzed measurements of the law of approach to ferromagnetic saturation. In addition these latter investigations give important information on the spin wave dispersion relation and the temperature dependence of the spontaneous magnetization. Our experimental results will be discussed within the framework of the theory of micromagnetism as already developed for amorphous materials [S-7]. It will be shown that the displacement of dws may be described successfully within the framework of potential theory.
2. Domain structures in amorphous alloys 2.1. General remarks For the study of domain structures in amorphous alloys the classical Bitter technique [8-l l] as well as the magnetooptical Kerr effect has been successfully applied [12-161. The first of these techniques may be applied to a study of surface domain structures of the as-quenched specimens whereas the latter technique generally requires a careful surface polishing. Therefore the Bitter technique is useful for a study of the effect of surface roughnesses on the dws whereas the Kerr technique is more appropriate for a study of the change of the dws under the influence of an applied magnetic field. As is well known domain structures are the result of a complicated energy balance between magnetostatic stray field energies, anisotropic magnetic energy terms, and the exchange energy usually contained in the domain wall energy. In soft magnetic materials the stray field energy &L,-,M~is much larger than anisotropy energies and the domain wall energy. The dws therefore is submitted to the condition of stray field
alloys
free spin configurations. This rather restrictive condition imposes the following for the dws: (1) disappearance of volume charges div M,(r) = 0, (2) disappearance
(1) of surface charges
n(r) - M,(r) = 0,
(2)
where n(r) corresponds to the unit vector perpendicular to the surface element at position r. Eq. (2) is fulfilled if the spontaneous magnetization at the surface is aligned parallel to the surface. Due to (1) and (2) the geometrical arrangement of domains is exclusively determined by magnetic anisotropy, the dw energy, and the shape of the specimen. 2.2. Anisotropy energies The domain patterns of amorphous alloys are determined by the following three different types of anisotropy energies: fie shape anisotropy. In order to reduce the magnetostatic stray field energy the spontaneous magnetization throughout the specimen tries to align parallel to the ribbon’s surface. Due to the smaller demagnetization field for a magnetization direction parallel to the ribbon’s axis the spontaneous magnetization prefers to be aligned parallel to the ribbon. The magnetoelastic coupling energy. In a material with non-vanishing isotropic magnetostriction, X,, internal stresses, 0, give rise to a magnetostrictive interaction energy
where cii corresponds to the ith component of the diagonalized stress tensor and -yi denotes the direction cosine of MS with respect to the ith coordinate axis. Structure anisotropy. The absence of long range order averages out those energy terms like the magnetic anisotropy in crystals. Nevertheless due to temperature gradients or magnetic fields occurring during the quenching process there may exist gradients in the distribution of metalloid atoms. Furthermore some atomic pair ordering of short range type (e.g., Fe-Ni pairs) may be induced due to gradients in the atomic distribution function.
H. Kronmiiller et al. /Magnetic properties of amorphous ferromagnetic alloys
55
In the following we present domain patterns where the above mentioned anisotropy energies play a dominant role. 2.3. Domain structures Our magneto-optical Kerr effect investigations were performed on amorphous NiFePB- and CoFeSiB-alloys as produced by the spin-quenching technique. The specimens were prepared by grinding with diamond paste and were then electrolytically polished in a solution of 4 nitric acid and $ methanol. After this treatment the specimens generally had a thickness of about 20 pm. In order to increase the optical contrast the polished specimens were evaporated with a A/4-i& of ZnS [ 171. Characteristic domain patterns observed in spinquenched alloys are shown in figs. l-6. 2.3.1. iQFe4oP14B6 (Metglas 2826) [13,1.5] In the general view given in fig. 1 for the domain structure for zero applied field three different types of domain patterns may be observed: (i) wide, wavy laminae with a width of 25 pm. In some regions these wide laminae show preferred orientations parallel or perpendicular to the ribbon axis. (ii) Patches of rather narrow, sometimes circular or zig-zag laminae with a domain width of ~3-5 pm. These patches have diameters of about 150-200 pm
Fig. 1. Domain structure of an asquenched narrow zig-zag laminae (H = 0).
Fe40Ni40P14B6-ribbon
Fig. 2. Domain structure of narrow laminae compressive stresses (H = 0, Metglas).
in a region
of
and in general are embraced by wide laminae (see fig. 2). (iii) Starlike domain patterns where wide laminae are flowing out from one centre (see fig. 3). After an annealing treatment for 20 h at 473 K the wide laminae were oriented perpendicular to the ribbon axis (rolling direction) and the number of domain patches was considerably reduced (see fig. 4). Under an applied magnetic field the domain
at zero-applied
field with regions
of wide starlike
domains
and
56
H. Kronmiiller
Fig. 3: Starlike domain pattern stresses (H = 0, Metglas).
et al. /Magnetic
properties of amorphous ferromagnetic
alloys
in a region of dilatational
patterns show characteristic changes. Wide domains vanish at such low fields as H = 10 Oe (800 A m-l) (see fig. 5) whereas the domain patches vanish at 20 Oe (1600 A m-l) if the magnetic field is applied perpendicular to the narrow laminae of the patches (fig. 6). In a magnetic field applied parallel to the narrow
Fig. 4. Domain
structure
after an annealing
Fig. 5. Change of the domain field H = 800 A m-t applied laminae (Metglas).
treatment
structure under a magnetic perpendicular to the narrow
of 20 h at 473 K (H = 0, Metglas).
H. KronmtiIler et al. /Magnetic properties of amorphous ferromagnetic alloys
51
laminae these laminae vanish at the rather high fields of 180 Oe (14 400 A m-r). 2.3.2. FexoBzo (Allied Chemical 2605) [16] As shown in fig. 7 in FesoBzo similar domain patterns are observed as in the case of Ni4eFe4ePr4B6, For an applied field of 22 Oe (1750 A m-r) the wide domains have vanished (fig. 8). Within the patches one system of narrow laminae increases in width whereas another system shrinks in width. The narrow laminae vanish at about 62 Oe (4900 A m-r). By an annealing treatment the domain patches as well as the number of wide domains is reduced considerably. In some regions only two domains are observed over the whole width of the ribbon.
Fig. 6. Domain structure with vanishing narrow laminae for an applied field of 1750 A m-l (Metglas).
2.3.3. Co70FepYiPB~6 (Vakuumschmelze Hanau) This amorphous Co-alloy is characterized by a vanishing magnetostriction constant. Characteristic domain patterns of the as-quenched specimen are
Fig. 7. Domain structure of an as-quenched Fe 80 Bzo-ribbon showing wide and narrow laminae at zero-applied field.
58
Fig. 8. Narrow H=4900Am-‘.
H. Kronmiiller et al. /Magnetic
and wide domains
of Fego&o
under
properties of amorphous ferromagnetic
the influence
of an applied
alloys
field; (a) H = 0 A m-l;
(b) H = 1750 A m-l;
(c)
H. Kronmtiller et al. /Magnetic
Fig. 9. Domain
structure
of an asquenched
properties of amorphous ferromagnetic
Co 70FegSigB16-ribbon
shown in fig. 9. Together with wide laminae oriented perpendicular to the ribbon axis we observe a stripe domain structure which appears to be superimposed in the wide domains. After an annealing treatment at 500 K the stripe domains have vanished and the wide
Fig. 10. Domain
structure
of a Co7OFegSiqB16-ribbon
showing
wide laminae
alloys
59
and weak stripe domains.
domains show a marked tendency to become wider and to align parallel to the ribbon axis (see fig. 10). Under an applied magnetic field oriented perpendicular to the stripe domains these vanish at about 8 Oe (650 A m-l). With a magnetic field applied
after annealing
for 27 h at 500-515
K.
60
H.
Kronmiilleret al. /Magneticpropertiesof amorphousferromagnericalloys
parallel to the stripe domains the wide laminae vanished at about 40 Oe (3200 A m-l) whereas the stripe domains still were present at 88 Oe (7000 A m-l).
3. Interpretation
relation lutl ut - loci u, = 0 .
(4)
By means of eq. (4) we now may derive a number of important properties of the magnetization curve and the domain structure.
of domain structures 3.2. Magnetoelastic coupling energy
3.1. Directions of easy magnetization The arrangement of domains in the as-quenched alloy is characterized by a composition of wide laminae and patches of narrow laminae. The fact that the narrow laminae react mainly on magnetic fields applied perpendicular to them shows that this domain pattern corresponds to a closure domain structure. A model of domain structures derived from these sug gestions is given in fig. 11. According to these results we assume that within the wide laminae the easy directions lie within the plane of the ribbon whereas within the patches the easy directions are oriented perpendicular to the ribbon. As discussed previously the anisotropy energy responsible for these easy directions is suggested to be the magnetoelastic coupling energy. According to eq. (3) in the case of a positive magnetostriction, tensile stresses produce an easy axis parallel to the tensile axis whereas compressive stresses lead to an easy axis perpendicular to the compression axis. From an elastomechanical point of view the stress axes should lie parallel to the ribbon. Therefore wide laminae are attributed to regions of tensile stresses, ut , and narrow laminae patches to regions of compressive stresses, uC. As shown by Albenga [ 181, the volume averages of internal stresses vanish. By introducing the partial volume fractions ut and uC of tensile and compressive stresses this leads to the following
For a random distribution of domains (($) = 3) the volume average of the magnetoelastic coupling energy is given by K, = $Qlatlut
+ IU&,) = &W,i%.
(5)
If no other anisotropy energies are present K, is directly measured by the magnetization energy. 3.3. Magnetization processes Due to the rather inhomogeneous domain structure of the as-quenched alloys the magnetization is controlled by at least three main types of magnetization processes. (i) In the low field region (zero internal field) magnetization takes place by 180”-dw displacements of wide domains. Accordingly the initial magnetization occurs in the regions of tensile stresses. (ii) The narrow laminae patches in a first stage are magnetized by a growing of the energetically favourable closure domains as indicated in fig. 6. A full magnetization of the closure domains is obtained for the magnetic field HI = Kc/J,,
(6)
where K, = ~Aslocl
(7)
corresponds to the anisotropy energy within the closure domains. (iii) In a second stage within the narrow laminae patches rotational processes take place as shown in fig. 12. It may happen that stage I and stage II take place simultaneously if the internal magnetic field increases at external fields smaller than HI. The spontaneous magnetization in the narrow laminae is aligned parallel to the applied magnetic field for Fig. 11. Schematic domain model showing wide laminae narrow domains with their closure structure.
and
HII = 2K,/J,.
(8)
H. Kronmdller
et al. /Magnetic
properties of amorphous ferromagnetic
H
-
Hr
b
Fig. 12. Magnetization process within narrow domains. (a) Growing of the closure domains at small fields (H < HI). (b) Rotation of the spontaneous magnetization within the narrow laminae for HI < H < HII.
3.4. Domain wall width of narrow domains 3.4.1. Ni40Fe40P14B20 and FegoBzo alloys The domain wall width of the stray-field free laminar domain structure of the Landau-Lifshitz is given by DO = (2TO~BfKc)“2.
type (9)
Here Tn = 4(AK,)l12 corresponds to the specific wall energy in the compressive regions and To denotes the thickness of the ribbon. The exchange constant, A, may be determined from the spinwave stiffness constant, D,, , as discussed in section 6. From (9) we obtain for the magnetic anisotropy energy, K,, within the closure domains and for Yn K, = 64AT;/D;,
(10)
TB = 32ATO/D&
(11)
The numerical
61
(lo), and (11) are given in table 1 for Ni4eFe4ePr4B6 and FeseB2e. From our results for K, we may determine the amplitude of the compressive stresses from eq. (7). With h, = 11 X 10e6 [ 191 we find for Ni40Pe4ePr4Be Us = 0.17 X 10’ N m-’ and for FeseB2e with h, = 30 X lop6 [22] we obtain u, = 1.1 X lo8 N rnp2. It should be noted that the internal stresses are found to correspond to -1% of the shear modulus.
a -
alloys
3.4.2. Co 70Fe5SigB16-a110ys Alloys of this composition are characterized by a vanishing magnetostriction. The domain structures in these cases are determined by shape and structural anisotropies. According to fig. 9 the domain structure is composed of wide domains and so-called stripe domains [20-221 with a wavelength of h - 6 pm. The appearance of a stripe domain configuration points to the fact that we are dealing with an oblique or perpendicular anisotropy which may be due to a structural anisotropy. As we may assume that the shape anisotropy is much larger than the oblique anisotropy, the spontaneous magnetization is tilted out of plane by only a small angle Oe. This leads to the configuration of wide domains shown in fig. 13. In the case of a homogeneous magnetization within the wide domains, surface charges would produce a large magnetostatic energy. In previous papers it has been shown [2 1,231 that these surface charges are suppressed by the formation of a stray-field free stripe domain structure. The conditions for the formation of such stripe domains in the case of a perpendicular anisotropy energy, KI, are the following:
values derived for K, and Yn from (7)
Table 1 Material parameters of the domain structure and of the domain walls in Metglas and FegoBZo. into SI-units the following relations hold: 1 Oe = (103/4n) A m-t, 1 G = lop4 T, 1 erg cm-l J m-‘, 1 erg cmm3 = l/10 J mv3
For a conversion = lo-’ J m-l;
of the cgs-units 1 erg cmP2 = 10e3
HI
47rM,
(Oe)
(G)
To (pm)
Do (pm)
A (erg cm-‘)
/(HI) KC‘(Do) (erg cmP3)
6B (crm)
YB (erg cme2)
Ni4oFe4oP14B2o
20
8000
20
4-6
3.1 x 10-7
0.62 X lo4 2.8 x 103
0.314
0.11
FesoB20
90
1600
20
3
5
6.0 4.9
0.10
0.64
x 10-7
X lo4 x 104
H. Kronmiiller et al. /Magnetic
62
properties of amorphous ferromagnetic
susceptibilities x0, large Rayleigh constants CY,and small coercive fields Hc. Fig. 14 presents the magnetization curve in the low field region and in fig. 15 the dependence of the quantity M/H on H is shown. From the linear dependence of M/H on H follows the validity of Rayleigh’s law. Previously the parameters x0, (Y,HG and H, were determined on the assumption that the displacement of the dw may be described by potential theory [24-271. Within the framework of this model it is assumed that the total force acting on a dw is given
Fig. 13. Model of wide domains and superimposed stripe domains in the case of a perpendicular easy direction.
To > 2n(A/K1)“’
by
where To corresponds to the thickness of the ribbon. In the case where K&p&) << 1 the wavelength of the stripe domain is given by XsD = 4n(A/K#‘*.
(12)
As in the case of an oblique anisotropy the only important term is K. sin* Oo which is connected with the component of MS perpendicular to the ribbon; we obtain X,,
= (4n/sin O,)(A/Ko)‘/*
(13)
.
The anisotropy constant K,, may be determined from the critical field for which the stripe domains and the wide domains vanish if applied perpendicular to the laminae. With H,, = 5 Oe we obtain from H,, = 2K,,/J, the anisotropy constant K. - 2.5 X lo3 erg cmA3 . With a reasonable value of A = 10m6 erg cm-’ we have to assume 00 = 20” in order to explain the observed wavelength of -6 pm.
4. The characteristic
alloys
parameters of the Rayleigh
P(Z)
= C
p(Z
-
Zi),
(19
i
where the sum extends over all individual pinning forces p(z - Zi) resulting from pinning centers at position Zi. The coordinate axis z is taken to be perpendicular to the dw. In general eq. (15) will lead to a fluctuating field of force and it may be shown [24,25] that the probability f(p) for the existence of a force between p and p t dp is given by the so-called normal distribution (see fig. 16) f(p) = (2nB)-1’2 exp(-p2/2@, where the correlation D/2
s
(16)
function B is defined as
cm}*dz.
(17)
-D/2
(Fn = dw area moving coherently, D = distance between neighbouring dws). By means of (16) the characteristic statistical properties of the field of force may be determined. Such properties are the
region
At small magnetic fields the magnetization obeys Rayleigh’s law M(H) = x,,H + d-f*
curve
(14)
where x0 denotes the initial susceptibility and OLthe Rayleigh constant. The magnetic field, HG , below which Rayleigh’s law holds in general is of the order of magnitude HG 5 OSH, where H, corresponds to the coercive field. Soft magnetic materials are characterized
1
18'
LTI 25 20 I5 -
0123456799
by large
Fig. 14. Magnetization (Fe40Ni40P14B6-ribbons)
10
curve of asquenched in the Rayleigh
&,H
Metglas region.
[IO-‘Tl
H. Kronmiiller et al. /Magnetic properties of amorphous ferromagnetic alloys
1 0
12
3
Fig. 15. The quantity the Rayleigh region.
4
5
J/p@
6
7
6
9
10
11
of Metglas as a function
poH CIO-‘TI
of H in
average value of the maxima of the force, the average wavelength, he, of P(z) and the average value, l/(dP/dz), of the reciprocal slope of P(z) at the dw positions where P(z) = 0 holds. Previously it has been shown that the potential theory leads to the following relations between x0, H, and QI(in Gaussian units):
xo& = Msvt -t x;/(olM,)
2
(ln
D/ho)l'* ,
= u,(37rll6)(ho/2D),
cuH,/xo = (g/3&)
(19)
[ln(D/~e)] ‘I2
(20)
In (18)-( 19) ho corresponds to the average wavelength of the field of force. From statistics it follows that ho - 46 n . All other quantities in the above relations x0, H,, CY,the partial volume vt of the tensile stresses and the laminae width D are known from
irrev. P(z)
Fig. 16. Definition of characteristic force acting on a domain wall.
parameters
of the field of
63
experiment. The partial volume vt appears in (18) and (19) because in the Rayleigh region dw movements take place only within regions of tensile stresses. In table 2 we give the experimental parameters of NiFeBalloys required for a test of eqs. (18)-(20), and fig. 17 shows OlH, as a function of x0. Because of the small variation of ln(D/ho) this function according to (20) should correspond to a straight line with a slope (g/3&) [ln(DlMl ‘I2 . The dashed line in fig. 17 corresponds to the slope obtained for D and 6n as determined for Fe40N&0P14B6 (see table 1). The agreement between experiment and theory as demonstrated by fig. 17 is surprisingly good for a statistical theory. The experimental points obtained for the CoFeSiB-alloys deviate considerably from the theoretical prediction thus indicating that in magnetostrictionless materials other, probably surface effects, are dominant for the magnetization processes.
5. Law of approach to ferromagnetic
saturation
5.1. General remarks Whereas domain patterns are suitable for a study of long range stresses with wavelengths h > 5 pm short range stresses with wavelengths 10 A < h < 1 pm may be studied by means of the field dependence of the high-field susceptibility. The theoretical background of this method has been developed in numerous papers [28-331, [S-7]. In amorphous alloys the spin moments are not aligned parallel to the internal magnetic field because of the following three perturbations: (i) as a consequence of the random distribution of atoms in ideal amorphous alloys with Bernal structure, intrinsic material properties, such as the local spin-orbit and exchange couplings as well as the modulus of the spontaneous magnetization, IM(r)l fluctuate randomly. Due to these intrinsic fluctuations the orientation of the spontaneous magnetization M(r) deviates in general from the direction of the internal magnetic field. Previous theoretical investigations [7], however, have shown that the effect of these intrinsic fluctuations on the law of approach to saturation is negligible in the case of transition metal-metalloid alloys. The same result holds for the influence of non-magnetic precipitates [34]. (ii) In real amorphous alloys there exist structural
H, Kronmiiller et al. /Magnetic
64
properties of amorphous ferromagnetic
alloys
Table 2 Characteristic parameters of the Rayleigh region of amorphous alloys. The material parameters are given in “Gaussian” units in the upper rows and in SI units in the lower rows. In order to convert the material parameters from “Gaussian” units into SI units, Js has to be multiplied by 4n X 10e4, H, by no X 10e4, x0 by 4n, and oi by 4n X lo4 0.Q = 1.256 x lo6 (V s A-’ m-l) (kc) Js (I’)
Hc n@c
(‘I)
/ /
t
(G 0eC2)
O1(T-l)
l-l
4.8 x 10’ 6.03 x lo2
3.16 X lo3 3.97 x 108
1.273 16.00 x 10-l
1.32 x 10-l 1.32 x 1O-5
9 x 10’ 1.13 x 103
2 x 103 2.51 X lo8
1.034 13.00x
1.08 x 10-l 1.08 x 1O-5
9.3 x 101 1.17 x 103
2.85 X lo3 3.58 X lo8
0.915 11.50 x 10-r
8.6 8.6
x 10-2 x 10-e
1.09 x 102 1.37 x 103
2.5 X lo3 3.14 x 108
0.796 10.00 x 10-l
6.5 6.5
x 1O-2 x lO-‘j
1.13 x 102 1.42 x lo3
2.74 X lo3 3.44 x 108
0.6685 8.40 x 10-l
4.25 x 1O-2 4.25 x 1O-6
1.2 x 102 1.51 x 103
6.58 X lo3 8.27 x lo8
0.43 5.40 x 10-r
4.9s x 10-Z 4.95 x 10-e
1.3 x 102 1.63 x lo3
7.69 x lo3 9.66 x lo8
0.533 6.70 X 10-l
1.98 x 1O-2 1.98 x lo@
2.5 x lo2 3.14 x 103
1.6 x lo4 2.01 x 109
0.482 6.06 x 10-r
4.29 x 10-Z 4.29 x 10-e
4.3 x 102 5.40 x 103
2.6 x lo3 3.27 x lo8
0.433 5.44 x 10-l
9.24 x 10-Z 9.24 x 10-e
1.1 x 102 1.38 x lo3
4.1 x 103 5.15 x 108
10-r
deviations from the ideal Bernal structure giving rise to internal elastic stresses. Due to the magnetoelastic coupling energy these internal stress fields produce
200
xo
x 1O-2 x lo@
0.621 7.8 x 10-l
4.2 4.2
(G Oe-‘)
(Oe)
X F&O NiJo BZO
x FeroNlra 8x1
an inhomogeneous spin arrangement as shown qualitatively in fig. 18 for a dislocation dipole. (iii) In addition to static inhomogeneities at finite temperatures, T,dynamic spin inhomogeneities exist due to thermally excited spinwaves. Under the action of an applied field, H, the deviating spins rotate into the direction of the applied field. This rotation process is field dependent and may be described by the empirical law of approach to saturation:
J(T, H> = Js [ 1 - (T/To>~'~ 1 + AJpara alI2
-H’12-g-$’
Fig. 17. Test of the self-consistency relation (20) of the potential model for dw displacement in amorphous alloys Fego.-JVi,B2o and CoFeSiB-alloys (compare table 2).
al
a2
(T H)
(21)
where J, corresponds to the absolute saturation polarization J, = ~~~~ = J(0, -), To denotes a characteristic temperature, and the parameters ap describe the effects of the microstructure. The different terms
H. Kronmiiller et al. /Magnetic properties of amorphous ferromagnetic alloys
-
_\---,‘-+
--
-‘-_;
_
-
A---
field range we plot J(T, H) as a function of l/W. Figs. 19 and 20 show the results of such plots for spin-cooled amorphous alloys. In the case of a Fe40Ni4ePr4B6 alloy the approach to saturation is described by a H-’ -term for H < 2000 Oe. Somewhat more complicated is the behaviour of a Fe~eNLreB2e-alloy which in the field range 100 < H < 300 Oe obeys a l/H-law and in the field range 300
-
-
-_
65
-
I+
Fig. 18. Schematic representation of the spin arrangement in the neighbourhood of a dislocation dipole.
= J(T, HI -
AJp,
Js(T, 0) + q 3
where JS(T, 0) corresponds
2
(23)
to the spontaneous
polari-
in (2 1) have the following meaning:
(1) the first two terms correspond to the effect of thermally excited spinwaves and to the so-called paraprocess AJ,,,(T, H) [35]. (2) Point defects with stress fields varying as u a l/r3 give rise to the l/H’12-term. (3) Straight dislocation dipoles with stress fields u a l/r2 lead to a l/H-term if the distance Ddtp between the two dipole dislocations is smaller than the exchange length: Hdip <
Kfil
= (2~4/~cf~)“~e
(22)
(4) The l/H2 -term may have different origins. (i) Single dislocations with stress fields varying as u a l/r or dislocation dipoles with Ddip > Kfil. (ii) Magnetic anisotropy due to long range stresses with wavelength h >pm. 5.2. Experimental
results
ECq.(21) shows that a determination ofJ,(T, 0) = Js [ 1 - (7’/re)3/2 ] for zero field in general is rather diffvult because in the high field region the para becomes important whereas at lower effect AJ,, fields the microstructural effects of the type H-” are prevailing. Therefore different methods are used in order to analyse these two field ranges. In order to determine the microstructural terms in the lower
1
0
C
20
LO
60
f&H
CT-“21
Fig. 19. The magnetization curves of Fe40Ni40Pt4B6 for different temperatures as a function of l/&H) [ 361.
H. Kronmtiller et al. /Magnetic
66
properties of amorphous ferromagnetic
alloys
JCTI
1.20 16 t I.16
0.4 1.12 I
0
200
100
I
300
Fig. 20. The magnetization curves of Fe46Ni4uB2u ent temperatures as a function of l/(r&12.
Ip_Fil'[T'l
for differ-
zation obtained by the extrapolation of l/H” --f 0, and the third term corresponds to the correction of the microstructural effects (compare insert in fig. 19). According to Holstein-Primakoff [35] the para effect may be written as J para = c(n
(24)
fHP
with the Holstein-Primakoff
0123456789
I
4oo
IO
f"p [T”21
Fig. 22. The para effect hlpara = n&Mpara for Fe4uNi4uB20 represented as a function of the Holstein-Primakoff function fHp for different temperatures.
cases the para effect is found to depend linearly on fHp. From the slopes of the straight lines we obtained the functions C(7) as represented in fig. 23. For temperatures T < 200 K a linear temperature dependence was found and the coefficients c = dC/dT are given by c = 1.3 X lo-’ for Fe40NLroP14B6 and c = 0.89 X 10m5 for Fe40Ni40B20 in units of T112 K-‘.
function 5.3. Interpretation of microstructural effects
and the temperature function C(T) proportional to T. In fig. 21 and fig. 22 the para effect AJ,,, is represented as a function of fHp for both alloys. In both
.a2
As discussed in section 5.1 internal stresses varying according to l/r give rise to characteristic field dependences in powers of l/H. The experimentally determined exponent of the 1/E-I”I-law therefore indicates which kind of internal stresses exist in amorphous alloys. Previous investigations of the effect of internal
295 K .10-3
5-
LFe40 Wo PM 6%
..‘-..-.x-x OY’IQ 0
2.0
L.0
91 K
/d /
)
Ix-
u I 6.0
1 8.0
L.2K I fH1?[T"*l 1
.3g. 21. The para effect Upara = n&Vfpara for Fe4ONi40Pt4Be represented as a function of the Holstein-Primakoff function fHp for different temperatures.
0
I
A0
1
80
Fig. 23. The temperature and Fe40Ni40BZO.
I
120
function
I
160
I
200
1
240
I
T CKI
C(T) for Fe4uNi4uP14Be
H. Kronmtiller et al. /Magnetic properties of amorphous ferromagnetic alloys
stresses on the law of approach to saturation were based on the concept of lattice dislocations with their strength characterized by a discrete Burgers-vector b and a well-defined line direction. Obviously this concept cannot be applied to amorphous materials without any modification. Here it is referred to the work of Krijner [37 ] who has shown that any internal stress state can be represented by quasi dislocations, i.e., in this generalized continuum theory the concept of discrete Burgers vectors as in crystal lattices is replaced by the concept of a dislocation density where the effective strength, b,rf, of a quasi-dislocation may correspond to a “Burgers vector” much less than the nearest neighbour atomic distances. Quasi-dislocations in amorphous materials may result from mass density fluctuations which were quenched-in during the rapid cooling process. A simple two-dimensional model of such a mass density defect producing dislocation stresses is shown in fig. 24. Here it was assumed that vacancy-like defects have agglomerated during the quenching process thus forming a diluted zone which leads to a local collapse of the lattice in these regions. This lattice contraction induces elastic deformations in the surrounding amorphous structure. Another type of stress sources may result from planar precipitates of metalloid atoms which induce compressive stresses. This type of structural inhomogeneities corresponds to the counterpart of the vacancy type precipitations. Both types of defects may be considered as dipoles of quasi-dislocations with a stress field varying as l/r’. In section 5.1
it was discussed that stress fields of this kind in fact give rise to a l/Z&term in the law of approach to saturation if the stress centers forming the dipoles have a distance, Ddip, smaller than the exchange length KE’ . The general relation for the AZ-effect due to quasidislocation dipoles of density Ndtp is given by [41]
(26) where F= 1.67 denotes a numerical factor which takes into account the statistical distribution of orientations of quasi-dipoles. The elastic properties of the amorphous material are characterized by the shear modulus G and Poisson’s constant V. In the case of Metglas Fe40Ni40Pr4B6 we have G = 4.93 X 101’ erg cmm3 E 4.93 X 10” Jme3, and v = 0.37, ref. [38]. In (26) F. takes care of the effect of one isolated quasi-dislocation dipole, and is given by FO = M&KHDdip)
’ $(KHDdip)
+Ko(K~Ddip) K1 (KHDdip)
+ 70 -
0.51,
(27)
where y. = 0.5772 corresponds to Euler’s constant, denotes the exchange length given by (22), and K. and K1 denote the modified Bessel functions of the second kind and of order 0 and 1. The function Fi:,nt takes into account the interaction between neighbouring dipoles. If RiZ corresponds to the distance between the ith and jth dipole and the condition Rij > Ddip holds Fint may be written as KH
Fint = C i
eij(KO(KHRij)[ZO(KHDdip)
KHDdip
--Il(KHDdip)
2
+F
Fig. 24. Schematic model for the formation of quasidislocation dipoles in amorphous alloys by the agglomeration of vacancy-type point defects in planar regions.
61
-
11
Kl(KHRij)[Io(KHDdip)
- 11)
(28)
where eij = 1 for dipoles with the same polarity, and Eij = -1 for dipoles of opposite polarity. IO and Zr denote the modified Bessel functions of the first kind. The functions F. and Fint were calculated as a function Of the parameters KHDdip and KHRii for a square lattice of lattice constant DdiD, i.e., there are
H. Kronmiiller et al. /Magnetic properties of amorphousferromagnetic alloys
68
four nearest neighbours at a distance (Rii)r = Ddip, four next nearest neighbours at (Rii)z = d2 Ddip, etc. We furthermore assume that there is a strong correlation in orientation and polarity of all dipoles up to the nth shell of the dipole lattice. Under the assumption that all dipoles are due to an agglomeration of vacancytype defects we have to choose in (28) eij = 1. In fig. 25 the functions Fo + Fint and Fo are represented for n = 11 which corresponds to no = 60 neighbours. The function FO + Fint may be described by the following three regions: (i) region I: KHDdip < 0.12 m
FO +Fint
1
AJ=----
-
$KhDitp(l
9 FG’hzNdipb&
477 64
(1 - v)~A
(ii) region II:
AJ=-
X
region
111:
1.22
0.12
D&p(l +“o)$
(29)
Fig. 25. The functions ln(K@dip), calculated with no = 60 dipoles.
1.95
20.1
X+, h,,p
FO and FO + Fint in dependence for a square lattice
of dipole
of clusters
1.22 < KHDdip < 5
9 FG2X2N s dip b= eff 1.6 _l_ 4 (1 - V)%l!fS H2 ’
(iii)
AJ=-
+ no),
(30)
KHDdip > 20.
9 FG~~=N. s drp b2eff i 4 (1 - v)=& 9
{hl(iKHDdip)+ 70
-
OS}
.
(iii) At very large magnetic fields, in region III, between dipoles becomes negligible and AJ is due to isolated dipoles. For larger values of (Rtj)l and smaller values of 12this result holds for smaller vdues of KHDdip . For a test of the above results we have represented in fig. 26 the experimental quantity AJH’ which should lead to a function linear in H for small values of H and a constant for larger values of H. Just this prediction in fact is observed for FedoNiQoBzo. A quantitative calculation leads to the following parameters (using the same values for G and v as for Metglas): KHDdip > 20, the interaction
(31)
It should be noted that the field range of validity of the l/H-power law increases with decreasing numbers n of shells considered in Fint . The range of validity is large for (Rtj)l - Ddip and (Rij)l + 00, but is reduced for all other values of (Rtj)l . On the other hand the results for regions II and III remain nearly unchanged if we choose n > 11 whereas in region I AJ is proportional to the number of dipoles no within the dipole cluster, The main conclusion to be drawn from these results are (i) in region I we obtain the well-known l/Hdependence of dislocation dipoles (see insert in fig. 25). (ii) In region II the magnetic interaction term leads to an enhanced approach to the l/Hz -dependence. The transition from a l/H- to a l/H2-law takes place in the interval 0.12 < KHDdip < 1.22.
Ddip = 245 A, Ndip = l/Dzip = 1.8 X 10” b eff = 1.9 A,
and
cm -2 ,
n = 11.
In the case of Fe40Ni40P14B6 within the whole field range /..L,,H < 0.2 T a linear increase of AJH2 is observed. This result is in agreement with measurements of Kazama and Kameda [39]. In order to increase the field range within which the l/H-law holds we assume that in Metglas a correlation only exists between nearest neighbour dipoles (n = 1). In this case the l/H-law is valid up to KHDdip = 0.6, and we obtain as an upper limit of the dipole width Ddlp < 0.6 KE1 @OH = 0.2 T) = 35.4 a. This corre-
69
H. Kronrmiller et al. /Magnetic properties of amorphous ferromagnetic alloys
1251.0
0.9 1.15 -
0
0
0.01
0.02
0.03
0.04
0.05
0.06
Fig. 26. The quantity aJH2 as a function of rfl
0117#fCTl for
1CQO
2000
3000
4000
Fig. 27. The temperature dependence of the spontaneous polarization in Fe40N&P14B6 and Fe40NiaoB20 alloys.
Fe4oNkoB2o.
sponds to a dipole density Ndip = l/Diin = 8 X 10” -‘. With b eff = 2.5 A we then are able to reprolice the experimentally observed values of AJ. Obviously in the Fe40Ni40P14B,s-alloy investigated Ddi,, and the correlation effects are smaller than in Fe40Ni40B20 : probably because of different quenching rates during the quenching procedure. In fact the observed domain structures support this assumption because in Fe40Ni40B20 we have found nearly no stress-induced domain pattern. Furthermore we observe in annealed Fe40Ni40Pr4B6 a l/H2 term, too, which is attributed to increased dipole widths and a larger correlation between neighbouring dipoles. According to these results a self-consistent interpretation of the law of approach to saturation seems to be possible on the basis of stress sources which exert internal stresses similar to those of straight dislocation dipoles. The fact that the dislocation strength b,ff is found to be of the order of magnitude of the interatomic distances strongly supports this interpretation. Finally it should be noted that our experimental result of the transitions from a l/H to a l/H2 law is supported by recent investigations of the magnetic small-angle neutron scattering in Fe40Ni40PloCio-alloys by Gijltz [40]. 5.4. Temperature dependence of the spontaneous polarization A comparison of the temperature dependence of J, with the theory of thermal spinwaves requires a careful determination of J, for zero applied field. As outlined in section 5.2 the spontaneous magnetiza-
tion J,(T, 0) for H = 0 can be determined by an extrapolation of the law of approach to saturation from the lower field range into the high field range. According to figs. 19-20, we obtain J, (T, 0) from the intersections of the l/H”-plots ofJ(T, H) with the coordinate axis at l/H” + 0. By this method we eliminate the microstructural term as well as the para effect of the spinwaves. The results of such an extrapolation are shown in fig. 27 where J,(T, 0) is plotted versus T312. In both alloys a T 3/2-law was found to be valid up to one half of the Curie temperature. Taking into account our results for the spinwave para effect we obtain for the field and temperature dependence of J,:
J&T H>= Js 11- (T/To)~‘~1 + cTfHp,
(32)
where fHp corresponds to the Holstein-Primakoff function [35]. The experimental parameters To and c are summarized in table 3. As is well known Bloch’s T3/2-law holds if the spinwave energy spectrum, ek, obeys a square dependence ek = D,,k'
,
(33)
where D,, corresponds to the so-called spinwave stiffness constant and k denotes the spinwave wavenumber. The validity of Bloch’s T312-law up to high temperatures can only be understood if the k4 terms in ek play a minor role due to a high density of states for the low energy spinwaves. The spinwave stiffness constant D,, is related to the parameter TO of Bloch’s law by
Ds,
=-& [Z~~;;B]2’3kB~o,
H. Kronmiiller et al. /Magnetic
70 Table 3 Material constants
of amorphous To (K)
F’QoNi4oPldb FedGoB
FegoBZo
]161
properties of amorphous ferromagnetic
alloys D SP (meV A2)
840
90
1300
191
1270
91
and the exchange constant A is given by A =Ms(%
~)&&‘~B
with g corresponding to the g-factor. Our results for To, D,, and A are given in table 3. References [l] H.-R. Hilzinger, A. Mager and H. Warlimont,
[2]
[3] [4] [S]
[6] [7] [8] [9]
[lo] [ 1 l] [ 121 [13] [ 141
[ 151 [16]
alloys
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A (J m-‘)
c
(T=OK)
@/K)
3.1
1.3
x 10-12
x 10-S
8.07 x lo-l2
0.89 X 1O-5
5
_
x 10-12
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