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Experimental approach to coercivity analysis in hard magnetic materials
Journal of Magnetism North-Holland
and Magnetic
Materials
83 (1990) 183-188
183
EXPERIMENTAL APPROACH TO COERCIVITY IN HARD MAGNETIC MATERIALS D. GIVORD,
Q. LU, M.F. ROSSIGNOL,
ANALYSIS
P. TENAUD
and T. VIADIEU
Luboratoire Louis Neel, 166 X, 38042 Grenoble Cedex, France Different processes may determine magnetization reversal in hard magnetic materials. The conditions under which these occur and their exact mechanisms are not well understood. In this paper some of the magnetic measurements which appear to be most meaningful are described and their implications with respect to physical processes involved are discussed. Whatever the exact coercivity mechanism is, the observed behavior is discussed using the concept of activation volume in which magnetization reversal is initiated. When the size of this volume is of the order of the size of the magnetic particles, it is suggested that collective magnetization processes are involved. The activation volume is on the contrary much smaller than magnetic particles and varies with temperature approximately as a3 when domain wall nucleation and propagation occur. The coercive field is deduced to be weak with respect to the anisotropy in the activation volume. The sign and the strength of dipolar interactions are discussed. 1.
Introduction
There are two main areas for the applications of hard magnetic materials, namely recording and permanent magnets. The characteristic property of these materials, coercivity, is due to some well-defined intrinsic magnetic properties in a material exhibiting a peculiar microstructure. For instance, in particulate media for recording, the coercivity comes form the ferro- or ferrimagnetic character combined with very small individual particles. In sintered magnets, it comes from the high anisotropy of crystallites embedded in an essentially non-magnetic matrix. These different microstructures have in common their heterogeneous character but the necessary interplay between microstructure and magnetic properties for the existence of coercivity in a given material is in general still poorly understood. From a theoretical point of view, the simplest approach to magnetization reversal processes is due to Stoner and Wohlfarth [l], it assumes coherent rotation of the magnetization. In a spheroid whose major axis coincides with the easy axis of magnetocrystalline anisotropy, the coercive field is:
where K, is the first-order anisotropy contant, M, the spontaneous magnetization, N,, and N, the demagnetizing factors along the major and minor axes. However, it appears that this process almost never occurs in reality, and the coercive field measured experimentally is always significantly smaller than that given by relation (1). The experimental processes which may occur can be classified into two broad categories. i) Collective processes in which magnetization reversal involves simultaneously all the magnetic moments of the specimen. These may take place in small particles and when 0304-8853/90/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
the magnetocrystalline anisotropy is low. In a prolate spheroid which is aimed to represent elongated particles, magnetization curling is expected to occur [2,3]. This process allows the energy associated with shape anisotropy to be decreased at the expense of exchange interactions. The nucleation field, assimilated in general with the coercive field is [4]: H, = 2K,/~,94,
- N,,Ms + (%/2S*)k,
(2)
where S = m x 10-3MsA-“2R, R being the semiminor axis, A the exchange constant, k a parameter dependent on the sample shape and whose value is of the order of 1. ii) Non-collective processes in which magnetization is inverted by domain wall nucleation and propagation. These processes are important in materials with high anisotropy. They allow the magnetization to be reversed without having to pay the very large anisotropy energy. Yet, as was shown originally by Brown [5], an energy barrier should theoretically prevent nucleation from occurring. However, in a real material, defects are always present, in particular at the sample surface. They act as seeds for nucleation of domain walls [6]. There is no well-developed theory to describe these phenomena. By analogy with relations (1) and (2), a more or less empirical formula is often used to express the coercive field:
where a and N,,, are two parameters representing the relative importance of magnetocrystalline anisotropy and dipolar interactions, respectively. The occurrence of one or the other of the magnetization processes described above and the magnetic and microstructural parameters relevant in this respect, are very difficult to determine from an experimental point of view. The primary reason is that magnetization rever-
184
D. Giuord et al. / Coerciuity analysis in hard magnetic materials
sal cannot be directly observed. It has a transitory character and generally takes place in a very small part of the considered specimen. An experimental approach necessarily relies on an analysis of the effects of different physical parameters on the value of H,. In this paper, we describe some of the magnetic measurements which appear to be most meaningful and try to delineate their exact implication with respect to physical processes involved.
00
200
460
600
1000
2. Qualitative discussion of coercivity mechanisms McCunie
In certain hard magnetic materials, the initial susceptibility of thermally demagnetized samples is low. indicating that no free domain wall displacement takes place. This property means either that crystallite size is below the critical size for single-domain particles or that domain walls are pinned. Microstructural studies may allow one to select between these possibilities. The former applies to particles for recording ( y-Fe,O,, CrO,), the latter to Sm (Co-Cu), magnets and Gd-Fe amorphous films for magneto-optic recording. In the case of Nd-Fe-B MQ magnets prepared from melt-spun ribbons, it has been suggested from the observation of domains whose size is larger than the one of individual particles, that domain-wall pinning is involved [7]. However, Griinefeld and Martinek [8] have pointed out that these domains could be determined by magnetostatic interactions between small crystallites and that magnetization reversal involving individual singledomain particles cannot therefore be excluded. In other materials such as sintered SmCo, or Nd-Fe-B magnets, a large initial susceptibility is measured on thermally demagnetized samples. This behaviour means that free domain wall displacement occurs. It indicates i) that the size of the individual crystallites of which the magnet is constituted is larger than the critical size for single-domain particles and ii) that no pinning of domain walls occurs inside the crystallites. Then, non-collective processes are a priori involved in magnetization reversal. Indeed, in case where the occurrence of such processes is established, their experimental analyses reveal that the initial volume of reversed magnetization is much smaller than the critical size for single-domain particles (see section 4). Once a domain of reversed magnetization is formed, it may expand until it occupies the whole volume of the considered crystallite. In this case, coercivity is really determined by the process of domain nucleation. It may also be that the above mentioned domain remains in the vicinity of the grain surface where it is usually formed and expands only when the field reaches the domain wall propagation field. The domain of reversed magnetization is so small that it cannot, in general, be detected
00
50 D
100 Cm)
Fig. I. The dependence of coercivity on particle diameter. Upper: Y-Fe@, (after Eagle et al. [9]); o: 6% Co, l : 4.2% CO, +: 3% co. ~ curling curve: down: SmCo, (after Strnat et al. and McCurrie et al. [IO]).
in magnetization measurements. As a result this process can only be separated from true nucleation with great difficulty. As shown in fig. 1, the coercivity of ferromagnetic particles is size dependent. In y-Fe,O, small particles. the observed dependence is in agreement with the one calculated for curling [9]. The increase in coercivity observed down to the smallest particle sizes indicates that no coherent rotation is involved, otherwise the coercive field would become size-independent below a certain critical size. For the RCo, particles, it is well established that domain wall nucleation takes place at defects which act as seeds for magnetization reversal. Since the size of the defects is much smaller than the particle size, the observed dependence of the coercive field on grain diameter is rather surprising (fig. 1) [lo]. It is usually argued that the number of defects per particle decreases as their size is decreased. This interpretation is not totally convincing since reducing a material into powder inevitably introduces new additional defects. 3. Temperature dependence of the coercive field In AlNiCo magnets on y-Fe,O, particles [9], the temperature dependence of H, can be approximately fitted to a relation similar to (2) indicating some collective processes. In other systems, non-collective processes are expected. The temperature dependence of the coercivity is shown for instance in fig. 2 for ferrites and Nd-Fe-B sintered magnets. The calculated curves have been obtained in this instance using a formalism slightly different from relation (3) [ll] (see section 4). In all
185
D. Givord et al. / Coercivity analysis in hard magnetic materials systems, a satisfactory fit to the temperature dependence of H, can be obtained. When the results are scaled to relation (3), the coefficient ar obtained is in general 0.1-0.2. Schematically, it represents the departure from coherent rotation model, for which (Y= 1. At first sight it is surprising that the temperature dependence of H, can be fitted to a term which is proportional to the bulk anisotropy. This indicates in fact that the anisotropy in the region where magnetization reversal starts is proportional to the bulk anisotropy. According to Callen and Callen [12], the temperature variation of K, in a uniaxial system is proportional to M,’ and hence determined solely by the strength of the exchange interactions. The observed behaviour implies therefore that these interactions are not reduced in the region of initial magnetization reversal. In these systems, the initial volume of reversed magnetization being much smaller than the crystallite size, the crystallite shape does not lead to a shape anisotropy. The dipolar interactions can then be approximated as a large field which helps to invert the magnetization. Values of the order of M, (N,,, = 1) are deduced from the analysis of H,(T). Such large values and even larger values [13] can be accounted for by considering the strongly heterogeneous character of the magnetization state. In Sm(Co, Cu, Fe, Zr),_, magnets where coercivity is determined by pinning, the coefficient (Y assumes similar values as in sintered magnets; the coefficient N,,, is found to be negative, i.e. the dipolar interactions enhance the coercive field. An interesting implication in this instance is that the coercive field vanishes at the Curie temperature whereas in NdFeB or SmCo, sintered magnets it vanishes below T,, when the dipolar field is
0.4
0.3
‘;‘ 2 2
0.2
Ez
0.1
0’
0
I
I
I
I
100
200
300
400
T
”
0
500
(K)
Fig. 2. The temperature dependence of coercivity of NdFeB powder sintered magnets and Ba-ferrite magnets. o and 0 experimental points, calculated curves.
.. !J-h
.
. \
100 200
0
300
T
400
500
600
(K)
Fig. 3. The temperature dependence of coercivity MQI magnets. calculated curve.
of NdFeB
large enough to provoke domain wall nucleation/propagation in the absence of any applied field. As suggested by NCel [14], to interpret negative values for N,,, it may be considered that the initial domain configuration minimizes the local divergence of the magnetization which is not zero due to the heterogeneous character of the system. The same analysis of the temperature dependence of H, as in sintered magnets may be applied to Nd-Fe-B MQ magnets prepared from melt-spun ribbons (fig. 3) [15]. The value of the parameter a is again similar as in other systems. The parameter N,,, is found to be negative in MQI and MQII magnets. This does not imply, in our opinion, that domain wall pinning occurs as in Sm(CoCu) materials. Considering the isotropic distribution of grain orientation in these magnets, very large local divergence of the magnetization may exist. It may then be that the material tends to be divided into regions in such a way that the orientation of the magnetization in the crystallites at the boundary between two regions minimizes dipolar interactions. 4. Magnetic
aftereffect
In a field opposed to the direction of the magnetization, a coercive material is in a metastable state. It tends to reach the state of minimum energy and, under the effect of thermal activation, the magnetization varies with time. A logarithmic dependence is in general observed [16]. One defines the viscosity: S = -dM/dln
t
and
S, = S/xirr,
(4),(5)
where x irT is the irreversible susceptibility measured at constant aM’“/at, Mirr being the irreversible magnetization [17].
186
D. Gioord et al. / Coerciuity analysis in hard magnetic materials
According to usual formalism for thermal activation effects, a certain relaxation time 7 is necessary to overcome an energy barrier E: 7 = 7” exp E/kT,
where r0 is the minimum characteristic time for magnetization reversal. Due to the exponential variation of r with respect to E, it results that for usual times of measurement, the energy barriers involved are approximately constant, of the order of 25kT. Comparing the change of magnetization with time to the change of magnetization resulting from an increment AH of the applied field, it is then deduced that the volume u of material involved in magnetization reversal is [17,18]: ”=
W/‘w+fJ(xirr/S)
= (kT/@f,)W&).
+ poNeffvM~
As it is observed
+ 25kT.
experimentally
that a domain
0
‘,?
.
0
100
I
I
200
300
400
500
T (K)
(6)
This quantity is an important parameter in coercive processes, it is termed the activation volume [19,20] in the following discussion. But, considering the numerous simplifications made in the above analysis, the value of u obtained should only be taken as an evaluation. The temperature dependence of the activation volume obtained in sintered ferrite, Nd-Fe-B MQI magnets and in Ba-ferrites for magnetic recording is shown in fig. 4. The size of the activation volume in all these systems is much lower than the size of the magnetic particles. This shows that magnetization reversal is not a collective process but involves domain wall nucleation and propagation. In ferrites and Nd-Fe-B MQI magnets, the magnetocrystalline anisotropy being known, it is possible to calculate the domain wall width 6. The activation volume u is of the order of 10S3, and the temperature dependences of u and S3 are similar. This is particularly striking in fig. 4, since u and S3 increase with temperature in Nd-Fe-B MQI magnets, but decrease in ferrite magnets. This result is important as it suggests that the non-uniform magnetization site which is created in the initial stage of magnetization reversal already resembles a domain wall. This behaviour may be understood by considering that a domain wall represents the non-uniform magnetization state leading to an inversion of the magnetic moment direction at the minimum energy cost. A simple approach to magnetization reversal may be derived from these results [ll]. An energy barrier E,, which characterizes the coercivity in the considered system is assumed to be overcome under the effect of the applied field, of the dipolar interactions and of thermal activation: E, = povh4,H,
0
(7)
wall is
-
15
6
2 r0
t
c
T (K) Fig. 4. The temperature dependence of activation volume. (a) NdFeB MQI magnets, 0: u, 0: 6.713’; (b) sintered ferrite magnets, 0: u, 0: 18.6~3~ and recording-used modified Ba ferrite. +: 0.
formed in the initial stage of magnetization reversal, may be assumed to be proportional to y’, i.e.: E, - Y’S,
(8)
where y’ is the energy of the domain wall formed the surface of the activation volume. Two simple additional hypotheses are made is assumed to be proportional to u’/~. and y’ sumed to be proportional to y, the domain wall in the unperturbed hard magnetic material. It that E, = ayv2/’
E,
and
H, = a( Y/P&‘,~“~)
and s [ll]: s is asenergy results
(9) - N,,,W
- (25kT,‘/+W,).
(10)
The first term in relation (10) is generally dominant. y is proportional to w; from the results above, ui” is proportional to 6 and thus to m. It results that H, should be proportional to K,, in agreement with relation (3). Furthermore, as u is proportional to 6’, H, is deduced to be proportional to (l/~)~/‘. This relation-
187
D. Givord et al. / Coercivity analysis in hard magnetic materials
ship was experimentally found by Barbier for coercive field values varying over four orders of magnitude [16]. It is approximately obeyed for the three materials for which activation volumes are plotted in fig. 4. In y-Fe,O,, the activation volume at 300 K deduced from the value of S, is approximately 1-5 X lo-” cm3. It is of the same order of magnitude as the particle size. This behaviour is naturally expected for collective magnetization processes and tends to confirm that magnetization curling occurs. It would be very meaningful to determine the thermal variation of the activation volume in this system. In the case where collective processes are actually involved, u amounts to the whole size of the particle and should thus be temperature independent.
Let h,(O) be the coercive field measured in a small crystallite characterized by the anisotropy field HA (2K,/pOMs) along a direction antiparallel to the easy c-axis. In a field applied at an angle 8 from the c-axis the value is h,(B). The angular dependence of the coercive field is represented by the function g( 0): =h,(O)g(~).
(11)
In GdCo, [21] and BaO-6Fe,O, [22] single crystals, it is found that g(B) = l/cos 0, i.e. the coercive field increases very strongly as 0 is increased (fig. 5). In other hard materials for recording (Co-Cr, Co-y-Fe,O,) [23] or permanent magnets (NdFeB) [24,25], the coercive
1.0 IO
10
20
30
40
50
60
70
80
I
I
10
20
I
30
I
I
40
50
I
I
I
I
60
70
80
90
e
Fig. 6. Some other results on the angular variation of coercivity. 1: Co-Cr; 3: Ba-ferrite; 4: Co-y-Fe,O, (after Kubo et al. [22]); 2: Nd,,Fe,,B, magnets, our results.
5. Angular dependence of the coercive field
h,(B)
I o
90
0
e
Fig. 5. The angular variation of coercivity in the Stoner-Wohlfarth model and according to the l/cos 8 law, compared to the experimental results on GdCo, (0) (after Becker [21]) and on Ba-ferrite (0) single crystals (after Ratnam et al. [22]).
field H,(B) is also found to increase the variation is much less than l/cos For a small single crystal in which of the moments takes place (and thus function g(e) has been calculated Wohlfarth [I]: g(e)
=
l/[sin2’3
e +
COS*/~
ej3/*.
with 0, although t9 (fig. 6). coherent rotation h,(O) = HA), the by Stoner and
(12)
As 0 varies between 0 and a/2, the coercive field decreases from HA to HA/2 and increases again up to the initial value (fig. 5). This behaviour is in disagreement with all experimental results, which confirms that coherent rotation does not occur. Turning to collective non-uniform processes, the angular dependence of h,( 0) has been calculated by Aharoni [26] for the case of magnetization curling. The coercive field increases as 0 is increased. In our opinion, the physical origin of this phenomenon is the following. Magnetization curling allows the magnetization of small particles to be inverted in a field which is much lower than HA. When the field is applied at a certain angle 8, a torque is exerted on the moments, but if the applied field H is much lower than HA, the deviation of the moments from the easy axis can be neglected. This is the case if H is of the order of the coercive field as determined by curling. The effect of the applied field is then equivalent to that of its projection along the easy axis, i.e. the moments feel a field of strength H cos 0. It results that the angular dependence of the coercive field follows approximately a l/cos 0 law. The angular variation of H,(B) in y-Fe,O, may be attributed to the above mechanism [23]. However, the experimental variation does not correspond exactly to the theoretical one. It may be that the actual process is more complex than the one calculated, and it must also be considered that real
188
D. Givord et al. / Coercioity analysis in hard magnetzc materials
materials consist of an assembly of small crystallites which are not perfectly oriented with respect to one another. This fact is expected to lead to a damping of the function g(O). In Nd-Fe-B magnets, Givord et al. [24] have shown that the experimental variation of the coercive field H,(B) was due to a l/cos 0 dependence in individual crystallites, convoluted with the distribution of grain orientation independently determined. To account for this behaviour, the same argument as above can be used. It is thus derived that the effective anisotropy in the activation volume is large compared to the energy brought by the applied field. This effective anisotropy is due to the magnetocrystalline anisotropy in u and to exchange coupling between the magnetization in u and the magnetization in the rest of the crystallite. The former of these terms is expected to be dominant. In Nd-Fe-B magnets, the coercive field is about l/10 the anisotropy field. It is then concluded that the anisotropy in the activation volume is not drastically reduced with respect to the anisotropy in the bulk. This result is important to be taken into account when trying to understand coercivity in permanent magnets in terms of domain walls nucleated at sample defects. In other Nd-Fe-B magnets, whose coercive field values are higher, Kronmtiller et al. [25] have found that the angular variation of the coercive field is intermediate between the Stoner-Wohlfarth behaviour and the l/cos 0 law. This behaviour can simply be understood by considering that the coercive field is no more negligible with respect to HA [24]. In Pr-Fe-B magnets, where coercivity in excess of 8 MA/m, of the order of HJ2, is measured below 100 K, the coercive field is found to be minimum around 0 = 7r/4, in agreement with the Stoner-Wohlfarth model [27]. This suggests that, as the ratio h.(O)/H, increases, magnetization reversal in the activation volume resembles more and more coherent rotation. References [I] E.C. Stoner and E.P. Wohlfarth, (London) A240 (1948) 599.
Phil. Trans.
Roy.
Sot.
[2] E.H. Frei. S. Shtrikman and D. Treves, Phys. Rev. 106 (1957) 446. [3] A. Aharoni and S. Shtrikman, Phys. Rev. 109 (1958) 1522. [4] A. Aharoni, IEEE Trans. Magn. MAG-22 (1986) 478. [5] W.F. Brown, Jr., Rev. Mod. Phys. 17 (1945) 15. [6] G.W. Rathenau, J. Smit and A.L. Stuyts, Z. Phys. 133 (1952) 250. [7] T. Viadieu, Thesis, Universite Joseph Fourier, Grenoble 1 (1988). [S] M. Gronefeld and G. Martinek, private communication. [9] D.F. Eagle and J.C. Mallinson, J. Appl. Phys. 38 (1967) 995. [lo] K. Stmat, G. Hoffer, J. Olson and W. Ostertag, J. Appl. Phys. 38 (1967) 1001. R.A. McCurrie and G.P. Carswell, Phil. Mag. 23 (1971) 333. [ll] D. Givord, P. Tenaud and T. Viadieu, IEEE Trans. Magn. MAG-24 (1988) 1921. [12] E.R. Callen and H.B. Callen, J. Phys. Chem. Solids 16 (1960) 310. [13] E. Adler and P. Hamann, in: Proc. 4th Intern. Symp. on Magnetic Anisotropy and Coercivity in RE-transition Metal Alloys, ed. K.J. Strnat (Univ. Dayton. 1985) p. 747. [14] L. Neel, Ann. Univ. Grenoble 22 (1946) 299. [15] D. Givord, Q. Lu, M.F. Rossignol, P. Tenaud and T. Viadieu, to be published. [16] J.C. Barbier, Ann. de Phys. 9 (1954) 84. [17] R. Street. P.G. Melormick and Y. Estrein. in: Proc. 10th Intern. Workshop on Rare-earth Magnets and Their Application (Kyoto. 1989) p. 83. [18] R. Street and J.C. Wooley, Proc. Phys. Sot. A62 (1949) 562. [19] E.P. Wohlfarth, J. Phys. F 14 (1984) L 155. [20] D. Givord. A. Litnard, P. Tenaud and T. Viadieu. J. Magn. Magn. Mat. 67 (1987) L281. [21] J.J. Becker, in: AIP Conf. Proc. 5 (1971) 1067. [22] D.V. Ratnam and W.R. Buessem, J. Appl. Phys. 43 (1972) 1291. [23] 0. Kubo, T. Ido and H. Yokoyama, IEEE Trans. Magn. MAG-23 (1987) 3140. [24] D. Givord, P. Tenaud and T. Viadieu. J. Magn. Magn. Mat. 72 (1988) 247. [25] H. Kronmtiller, K.D. Durst and G. Martinek, J. Magn. Magn. Mat. 69 (1987) 149. [26] A. Aharoni. Phys. Stat. Sol. 16 (1966) 3. [27] M. Gronefeld and H. Kronmtiller, to be published.
and Magnetic
Materials
83 (1990) 183-188
183
EXPERIMENTAL APPROACH TO COERCIVITY IN HARD MAGNETIC MATERIALS D. GIVORD,
Q. LU, M.F. ROSSIGNOL,
ANALYSIS
P. TENAUD
and T. VIADIEU
Luboratoire Louis Neel, 166 X, 38042 Grenoble Cedex, France Different processes may determine magnetization reversal in hard magnetic materials. The conditions under which these occur and their exact mechanisms are not well understood. In this paper some of the magnetic measurements which appear to be most meaningful are described and their implications with respect to physical processes involved are discussed. Whatever the exact coercivity mechanism is, the observed behavior is discussed using the concept of activation volume in which magnetization reversal is initiated. When the size of this volume is of the order of the size of the magnetic particles, it is suggested that collective magnetization processes are involved. The activation volume is on the contrary much smaller than magnetic particles and varies with temperature approximately as a3 when domain wall nucleation and propagation occur. The coercive field is deduced to be weak with respect to the anisotropy in the activation volume. The sign and the strength of dipolar interactions are discussed. 1.
Introduction
There are two main areas for the applications of hard magnetic materials, namely recording and permanent magnets. The characteristic property of these materials, coercivity, is due to some well-defined intrinsic magnetic properties in a material exhibiting a peculiar microstructure. For instance, in particulate media for recording, the coercivity comes form the ferro- or ferrimagnetic character combined with very small individual particles. In sintered magnets, it comes from the high anisotropy of crystallites embedded in an essentially non-magnetic matrix. These different microstructures have in common their heterogeneous character but the necessary interplay between microstructure and magnetic properties for the existence of coercivity in a given material is in general still poorly understood. From a theoretical point of view, the simplest approach to magnetization reversal processes is due to Stoner and Wohlfarth [l], it assumes coherent rotation of the magnetization. In a spheroid whose major axis coincides with the easy axis of magnetocrystalline anisotropy, the coercive field is:
where K, is the first-order anisotropy contant, M, the spontaneous magnetization, N,, and N, the demagnetizing factors along the major and minor axes. However, it appears that this process almost never occurs in reality, and the coercive field measured experimentally is always significantly smaller than that given by relation (1). The experimental processes which may occur can be classified into two broad categories. i) Collective processes in which magnetization reversal involves simultaneously all the magnetic moments of the specimen. These may take place in small particles and when 0304-8853/90/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
the magnetocrystalline anisotropy is low. In a prolate spheroid which is aimed to represent elongated particles, magnetization curling is expected to occur [2,3]. This process allows the energy associated with shape anisotropy to be decreased at the expense of exchange interactions. The nucleation field, assimilated in general with the coercive field is [4]: H, = 2K,/~,94,
- N,,Ms + (%/2S*)k,
(2)
where S = m x 10-3MsA-“2R, R being the semiminor axis, A the exchange constant, k a parameter dependent on the sample shape and whose value is of the order of 1. ii) Non-collective processes in which magnetization is inverted by domain wall nucleation and propagation. These processes are important in materials with high anisotropy. They allow the magnetization to be reversed without having to pay the very large anisotropy energy. Yet, as was shown originally by Brown [5], an energy barrier should theoretically prevent nucleation from occurring. However, in a real material, defects are always present, in particular at the sample surface. They act as seeds for nucleation of domain walls [6]. There is no well-developed theory to describe these phenomena. By analogy with relations (1) and (2), a more or less empirical formula is often used to express the coercive field:
where a and N,,, are two parameters representing the relative importance of magnetocrystalline anisotropy and dipolar interactions, respectively. The occurrence of one or the other of the magnetization processes described above and the magnetic and microstructural parameters relevant in this respect, are very difficult to determine from an experimental point of view. The primary reason is that magnetization rever-
184
D. Giuord et al. / Coerciuity analysis in hard magnetic materials
sal cannot be directly observed. It has a transitory character and generally takes place in a very small part of the considered specimen. An experimental approach necessarily relies on an analysis of the effects of different physical parameters on the value of H,. In this paper, we describe some of the magnetic measurements which appear to be most meaningful and try to delineate their exact implication with respect to physical processes involved.
00
200
460
600
1000
2. Qualitative discussion of coercivity mechanisms McCunie
In certain hard magnetic materials, the initial susceptibility of thermally demagnetized samples is low. indicating that no free domain wall displacement takes place. This property means either that crystallite size is below the critical size for single-domain particles or that domain walls are pinned. Microstructural studies may allow one to select between these possibilities. The former applies to particles for recording ( y-Fe,O,, CrO,), the latter to Sm (Co-Cu), magnets and Gd-Fe amorphous films for magneto-optic recording. In the case of Nd-Fe-B MQ magnets prepared from melt-spun ribbons, it has been suggested from the observation of domains whose size is larger than the one of individual particles, that domain-wall pinning is involved [7]. However, Griinefeld and Martinek [8] have pointed out that these domains could be determined by magnetostatic interactions between small crystallites and that magnetization reversal involving individual singledomain particles cannot therefore be excluded. In other materials such as sintered SmCo, or Nd-Fe-B magnets, a large initial susceptibility is measured on thermally demagnetized samples. This behaviour means that free domain wall displacement occurs. It indicates i) that the size of the individual crystallites of which the magnet is constituted is larger than the critical size for single-domain particles and ii) that no pinning of domain walls occurs inside the crystallites. Then, non-collective processes are a priori involved in magnetization reversal. Indeed, in case where the occurrence of such processes is established, their experimental analyses reveal that the initial volume of reversed magnetization is much smaller than the critical size for single-domain particles (see section 4). Once a domain of reversed magnetization is formed, it may expand until it occupies the whole volume of the considered crystallite. In this case, coercivity is really determined by the process of domain nucleation. It may also be that the above mentioned domain remains in the vicinity of the grain surface where it is usually formed and expands only when the field reaches the domain wall propagation field. The domain of reversed magnetization is so small that it cannot, in general, be detected
00
50 D
100 Cm)
Fig. I. The dependence of coercivity on particle diameter. Upper: Y-Fe@, (after Eagle et al. [9]); o: 6% Co, l : 4.2% CO, +: 3% co. ~ curling curve: down: SmCo, (after Strnat et al. and McCurrie et al. [IO]).
in magnetization measurements. As a result this process can only be separated from true nucleation with great difficulty. As shown in fig. 1, the coercivity of ferromagnetic particles is size dependent. In y-Fe,O, small particles. the observed dependence is in agreement with the one calculated for curling [9]. The increase in coercivity observed down to the smallest particle sizes indicates that no coherent rotation is involved, otherwise the coercive field would become size-independent below a certain critical size. For the RCo, particles, it is well established that domain wall nucleation takes place at defects which act as seeds for magnetization reversal. Since the size of the defects is much smaller than the particle size, the observed dependence of the coercive field on grain diameter is rather surprising (fig. 1) [lo]. It is usually argued that the number of defects per particle decreases as their size is decreased. This interpretation is not totally convincing since reducing a material into powder inevitably introduces new additional defects. 3. Temperature dependence of the coercive field In AlNiCo magnets on y-Fe,O, particles [9], the temperature dependence of H, can be approximately fitted to a relation similar to (2) indicating some collective processes. In other systems, non-collective processes are expected. The temperature dependence of the coercivity is shown for instance in fig. 2 for ferrites and Nd-Fe-B sintered magnets. The calculated curves have been obtained in this instance using a formalism slightly different from relation (3) [ll] (see section 4). In all
185
D. Givord et al. / Coercivity analysis in hard magnetic materials systems, a satisfactory fit to the temperature dependence of H, can be obtained. When the results are scaled to relation (3), the coefficient ar obtained is in general 0.1-0.2. Schematically, it represents the departure from coherent rotation model, for which (Y= 1. At first sight it is surprising that the temperature dependence of H, can be fitted to a term which is proportional to the bulk anisotropy. This indicates in fact that the anisotropy in the region where magnetization reversal starts is proportional to the bulk anisotropy. According to Callen and Callen [12], the temperature variation of K, in a uniaxial system is proportional to M,’ and hence determined solely by the strength of the exchange interactions. The observed behaviour implies therefore that these interactions are not reduced in the region of initial magnetization reversal. In these systems, the initial volume of reversed magnetization being much smaller than the crystallite size, the crystallite shape does not lead to a shape anisotropy. The dipolar interactions can then be approximated as a large field which helps to invert the magnetization. Values of the order of M, (N,,, = 1) are deduced from the analysis of H,(T). Such large values and even larger values [13] can be accounted for by considering the strongly heterogeneous character of the magnetization state. In Sm(Co, Cu, Fe, Zr),_, magnets where coercivity is determined by pinning, the coefficient (Y assumes similar values as in sintered magnets; the coefficient N,,, is found to be negative, i.e. the dipolar interactions enhance the coercive field. An interesting implication in this instance is that the coercive field vanishes at the Curie temperature whereas in NdFeB or SmCo, sintered magnets it vanishes below T,, when the dipolar field is
0.4
0.3
‘;‘ 2 2
0.2
Ez
0.1
0’
0
I
I
I
I
100
200
300
400
T
”
0
500
(K)
Fig. 2. The temperature dependence of coercivity of NdFeB powder sintered magnets and Ba-ferrite magnets. o and 0 experimental points, calculated curves.
.. !J-h
.
. \
100 200
0
300
T
400
500
600
(K)
Fig. 3. The temperature dependence of coercivity MQI magnets. calculated curve.
of NdFeB
large enough to provoke domain wall nucleation/propagation in the absence of any applied field. As suggested by NCel [14], to interpret negative values for N,,, it may be considered that the initial domain configuration minimizes the local divergence of the magnetization which is not zero due to the heterogeneous character of the system. The same analysis of the temperature dependence of H, as in sintered magnets may be applied to Nd-Fe-B MQ magnets prepared from melt-spun ribbons (fig. 3) [15]. The value of the parameter a is again similar as in other systems. The parameter N,,, is found to be negative in MQI and MQII magnets. This does not imply, in our opinion, that domain wall pinning occurs as in Sm(CoCu) materials. Considering the isotropic distribution of grain orientation in these magnets, very large local divergence of the magnetization may exist. It may then be that the material tends to be divided into regions in such a way that the orientation of the magnetization in the crystallites at the boundary between two regions minimizes dipolar interactions. 4. Magnetic
aftereffect
In a field opposed to the direction of the magnetization, a coercive material is in a metastable state. It tends to reach the state of minimum energy and, under the effect of thermal activation, the magnetization varies with time. A logarithmic dependence is in general observed [16]. One defines the viscosity: S = -dM/dln
t
and
S, = S/xirr,
(4),(5)
where x irT is the irreversible susceptibility measured at constant aM’“/at, Mirr being the irreversible magnetization [17].
186
D. Gioord et al. / Coerciuity analysis in hard magnetic materials
According to usual formalism for thermal activation effects, a certain relaxation time 7 is necessary to overcome an energy barrier E: 7 = 7” exp E/kT,
where r0 is the minimum characteristic time for magnetization reversal. Due to the exponential variation of r with respect to E, it results that for usual times of measurement, the energy barriers involved are approximately constant, of the order of 25kT. Comparing the change of magnetization with time to the change of magnetization resulting from an increment AH of the applied field, it is then deduced that the volume u of material involved in magnetization reversal is [17,18]: ”=
W/‘w+fJ(xirr/S)
= (kT/@f,)W&).
+ poNeffvM~
As it is observed
+ 25kT.
experimentally
that a domain
0
‘,?
.
0
100
I
I
200
300
400
500
T (K)
(6)
This quantity is an important parameter in coercive processes, it is termed the activation volume [19,20] in the following discussion. But, considering the numerous simplifications made in the above analysis, the value of u obtained should only be taken as an evaluation. The temperature dependence of the activation volume obtained in sintered ferrite, Nd-Fe-B MQI magnets and in Ba-ferrites for magnetic recording is shown in fig. 4. The size of the activation volume in all these systems is much lower than the size of the magnetic particles. This shows that magnetization reversal is not a collective process but involves domain wall nucleation and propagation. In ferrites and Nd-Fe-B MQI magnets, the magnetocrystalline anisotropy being known, it is possible to calculate the domain wall width 6. The activation volume u is of the order of 10S3, and the temperature dependences of u and S3 are similar. This is particularly striking in fig. 4, since u and S3 increase with temperature in Nd-Fe-B MQI magnets, but decrease in ferrite magnets. This result is important as it suggests that the non-uniform magnetization site which is created in the initial stage of magnetization reversal already resembles a domain wall. This behaviour may be understood by considering that a domain wall represents the non-uniform magnetization state leading to an inversion of the magnetic moment direction at the minimum energy cost. A simple approach to magnetization reversal may be derived from these results [ll]. An energy barrier E,, which characterizes the coercivity in the considered system is assumed to be overcome under the effect of the applied field, of the dipolar interactions and of thermal activation: E, = povh4,H,
0
(7)
wall is
-
15
6
2 r0
t
c
T (K) Fig. 4. The temperature dependence of activation volume. (a) NdFeB MQI magnets, 0: u, 0: 6.713’; (b) sintered ferrite magnets, 0: u, 0: 18.6~3~ and recording-used modified Ba ferrite. +: 0.
formed in the initial stage of magnetization reversal, may be assumed to be proportional to y’, i.e.: E, - Y’S,
(8)
where y’ is the energy of the domain wall formed the surface of the activation volume. Two simple additional hypotheses are made is assumed to be proportional to u’/~. and y’ sumed to be proportional to y, the domain wall in the unperturbed hard magnetic material. It that E, = ayv2/’
E,
and
H, = a( Y/P&‘,~“~)
and s [ll]: s is asenergy results
(9) - N,,,W
- (25kT,‘/+W,).
(10)
The first term in relation (10) is generally dominant. y is proportional to w; from the results above, ui” is proportional to 6 and thus to m. It results that H, should be proportional to K,, in agreement with relation (3). Furthermore, as u is proportional to 6’, H, is deduced to be proportional to (l/~)~/‘. This relation-
187
D. Givord et al. / Coercivity analysis in hard magnetic materials
ship was experimentally found by Barbier for coercive field values varying over four orders of magnitude [16]. It is approximately obeyed for the three materials for which activation volumes are plotted in fig. 4. In y-Fe,O,, the activation volume at 300 K deduced from the value of S, is approximately 1-5 X lo-” cm3. It is of the same order of magnitude as the particle size. This behaviour is naturally expected for collective magnetization processes and tends to confirm that magnetization curling occurs. It would be very meaningful to determine the thermal variation of the activation volume in this system. In the case where collective processes are actually involved, u amounts to the whole size of the particle and should thus be temperature independent.
Let h,(O) be the coercive field measured in a small crystallite characterized by the anisotropy field HA (2K,/pOMs) along a direction antiparallel to the easy c-axis. In a field applied at an angle 8 from the c-axis the value is h,(B). The angular dependence of the coercive field is represented by the function g( 0): =h,(O)g(~).
(11)
In GdCo, [21] and BaO-6Fe,O, [22] single crystals, it is found that g(B) = l/cos 0, i.e. the coercive field increases very strongly as 0 is increased (fig. 5). In other hard materials for recording (Co-Cr, Co-y-Fe,O,) [23] or permanent magnets (NdFeB) [24,25], the coercive
1.0 IO
10
20
30
40
50
60
70
80
I
I
10
20
I
30
I
I
40
50
I
I
I
I
60
70
80
90
e
Fig. 6. Some other results on the angular variation of coercivity. 1: Co-Cr; 3: Ba-ferrite; 4: Co-y-Fe,O, (after Kubo et al. [22]); 2: Nd,,Fe,,B, magnets, our results.
5. Angular dependence of the coercive field
h,(B)
I o
90
0
e
Fig. 5. The angular variation of coercivity in the Stoner-Wohlfarth model and according to the l/cos 8 law, compared to the experimental results on GdCo, (0) (after Becker [21]) and on Ba-ferrite (0) single crystals (after Ratnam et al. [22]).
field H,(B) is also found to increase the variation is much less than l/cos For a small single crystal in which of the moments takes place (and thus function g(e) has been calculated Wohlfarth [I]: g(e)
=
l/[sin2’3
e +
COS*/~
ej3/*.
with 0, although t9 (fig. 6). coherent rotation h,(O) = HA), the by Stoner and
(12)
As 0 varies between 0 and a/2, the coercive field decreases from HA to HA/2 and increases again up to the initial value (fig. 5). This behaviour is in disagreement with all experimental results, which confirms that coherent rotation does not occur. Turning to collective non-uniform processes, the angular dependence of h,( 0) has been calculated by Aharoni [26] for the case of magnetization curling. The coercive field increases as 0 is increased. In our opinion, the physical origin of this phenomenon is the following. Magnetization curling allows the magnetization of small particles to be inverted in a field which is much lower than HA. When the field is applied at a certain angle 8, a torque is exerted on the moments, but if the applied field H is much lower than HA, the deviation of the moments from the easy axis can be neglected. This is the case if H is of the order of the coercive field as determined by curling. The effect of the applied field is then equivalent to that of its projection along the easy axis, i.e. the moments feel a field of strength H cos 0. It results that the angular dependence of the coercive field follows approximately a l/cos 0 law. The angular variation of H,(B) in y-Fe,O, may be attributed to the above mechanism [23]. However, the experimental variation does not correspond exactly to the theoretical one. It may be that the actual process is more complex than the one calculated, and it must also be considered that real
188
D. Givord et al. / Coercioity analysis in hard magnetzc materials
materials consist of an assembly of small crystallites which are not perfectly oriented with respect to one another. This fact is expected to lead to a damping of the function g(O). In Nd-Fe-B magnets, Givord et al. [24] have shown that the experimental variation of the coercive field H,(B) was due to a l/cos 0 dependence in individual crystallites, convoluted with the distribution of grain orientation independently determined. To account for this behaviour, the same argument as above can be used. It is thus derived that the effective anisotropy in the activation volume is large compared to the energy brought by the applied field. This effective anisotropy is due to the magnetocrystalline anisotropy in u and to exchange coupling between the magnetization in u and the magnetization in the rest of the crystallite. The former of these terms is expected to be dominant. In Nd-Fe-B magnets, the coercive field is about l/10 the anisotropy field. It is then concluded that the anisotropy in the activation volume is not drastically reduced with respect to the anisotropy in the bulk. This result is important to be taken into account when trying to understand coercivity in permanent magnets in terms of domain walls nucleated at sample defects. In other Nd-Fe-B magnets, whose coercive field values are higher, Kronmtiller et al. [25] have found that the angular variation of the coercive field is intermediate between the Stoner-Wohlfarth behaviour and the l/cos 0 law. This behaviour can simply be understood by considering that the coercive field is no more negligible with respect to HA [24]. In Pr-Fe-B magnets, where coercivity in excess of 8 MA/m, of the order of HJ2, is measured below 100 K, the coercive field is found to be minimum around 0 = 7r/4, in agreement with the Stoner-Wohlfarth model [27]. This suggests that, as the ratio h.(O)/H, increases, magnetization reversal in the activation volume resembles more and more coherent rotation. References [I] E.C. Stoner and E.P. Wohlfarth, (London) A240 (1948) 599.
Phil. Trans.
Roy.
Sot.
[2] E.H. Frei. S. Shtrikman and D. Treves, Phys. Rev. 106 (1957) 446. [3] A. Aharoni and S. Shtrikman, Phys. Rev. 109 (1958) 1522. [4] A. Aharoni, IEEE Trans. Magn. MAG-22 (1986) 478. [5] W.F. Brown, Jr., Rev. Mod. Phys. 17 (1945) 15. [6] G.W. Rathenau, J. Smit and A.L. Stuyts, Z. Phys. 133 (1952) 250. [7] T. Viadieu, Thesis, Universite Joseph Fourier, Grenoble 1 (1988). [S] M. Gronefeld and G. Martinek, private communication. [9] D.F. Eagle and J.C. Mallinson, J. Appl. Phys. 38 (1967) 995. [lo] K. Stmat, G. Hoffer, J. Olson and W. Ostertag, J. Appl. Phys. 38 (1967) 1001. R.A. McCurrie and G.P. Carswell, Phil. Mag. 23 (1971) 333. [ll] D. Givord, P. Tenaud and T. Viadieu, IEEE Trans. Magn. MAG-24 (1988) 1921. [12] E.R. Callen and H.B. Callen, J. Phys. Chem. Solids 16 (1960) 310. [13] E. Adler and P. Hamann, in: Proc. 4th Intern. Symp. on Magnetic Anisotropy and Coercivity in RE-transition Metal Alloys, ed. K.J. Strnat (Univ. Dayton. 1985) p. 747. [14] L. Neel, Ann. Univ. Grenoble 22 (1946) 299. [15] D. Givord, Q. Lu, M.F. Rossignol, P. Tenaud and T. Viadieu, to be published. [16] J.C. Barbier, Ann. de Phys. 9 (1954) 84. [17] R. Street. P.G. Melormick and Y. Estrein. in: Proc. 10th Intern. Workshop on Rare-earth Magnets and Their Application (Kyoto. 1989) p. 83. [18] R. Street and J.C. Wooley, Proc. Phys. Sot. A62 (1949) 562. [19] E.P. Wohlfarth, J. Phys. F 14 (1984) L 155. [20] D. Givord. A. Litnard, P. Tenaud and T. Viadieu. J. Magn. Magn. Mat. 67 (1987) L281. [21] J.J. Becker, in: AIP Conf. Proc. 5 (1971) 1067. [22] D.V. Ratnam and W.R. Buessem, J. Appl. Phys. 43 (1972) 1291. [23] 0. Kubo, T. Ido and H. Yokoyama, IEEE Trans. Magn. MAG-23 (1987) 3140. [24] D. Givord, P. Tenaud and T. Viadieu. J. Magn. Magn. Mat. 72 (1988) 247. [25] H. Kronmtiller, K.D. Durst and G. Martinek, J. Magn. Magn. Mat. 69 (1987) 149. [26] A. Aharoni. Phys. Stat. Sol. 16 (1966) 3. [27] M. Gronefeld and H. Kronmtiller, to be published.