Macroscopic Analysis of Magnetic Transitions in Anisotropic Magnetic Materials

The analysis of a macroscopical description, in terms of anisotropy constants, of some magnetic transitions in uniaxial systems is presented. The magnetic processes considered are the saturation along a hard direction and the first order magnetization process (FOMP). Evidence is given that a phenomenological description can offer some advantages in the physical interpretation of the mechanisms at the origin of the processes. The occurrence of FOMP is found to be consistent with the small angle canting model. Introduction of a microscopic theory. Furthermore a macroscopic Hard magnetic materials have being receiving increasing attention in the last few years due to their technological and thus economical importance. Magnetic recording materials and permanent magnet applications, which are the main uses of anisotropic ferromagnetic materials, are in great expansion. It is well known that technological potentialities greatly stimulate fundamental research and this has been particularly true for Nd-Fe-B type compounds [I], whose impact has made the current momentum in magnetic research particularly intense and promising. Among the many anisotropic magnetic compounds and alloys studied up to now, two classes of materials, namely hexaferrites and rare-earth intermetallic compounds, emerge for their peculiar magnetic properties. In spite of the differences in their crystal structure, chemical composition and magnetic interactions [2, 31, some relevant features such as spin reorientation transitions (SRT), first order field induced transitions and the presence of canted spin structures, are common to these two classes of hard magnetic materials [415] which are characterized by complex unit cells where different sites are available for different magnetic ions. This implies the presence of multiple contributions both to the exchange and to the anisotropy. The interplay between exchange and anisotropy, as well as the competition between different contributions to the magnetocrystalline anisotropy seems to be responsible for the complex phenomenology observed in these multisublattice systems, even if a general description has not yet been found. It is also remarkable that in all cases the main contribution to the magnetic anisotropy has a single ion origin. A phenomenological description can be given for the observed magnetization processes that in many cases, even neglecting the microscopic origin of the phenomena, allow for a simple and direct qualitative interpretation of the physical processes. Such a description could be considered propaedeutic to the formulation analysis, in terms of the anisotropy constants, alsoklows for a direct consistent use of the measured critical parameters such as anisotropy field, critical field and magnetization of a FOMP [4] and cone angle values. Additionally, the reliable measurement of critical parameters remains an important issue. The unique singular point detection technique (SPD) [16], has been proved to be an effective and reliable method to mesure both Ha and H,, directly in polycrystalline samples [17-201. Here an analysis of the macroscopical descriptions of some magnetic transitions in uniaxial systems is presented in an attempt to underline the usefulness of such treatments. The usefulness is mainly connected with the allowance of a straightforward qualitative analysis and the consequent possibility of a consistent comparison of different systems. Anisotropy field in the hard direction and its influence on coercivity The anisotropy field Ha can be technically defined as the field value needed to saturate a sample along a hard direction. Thus the approach to saturation along a hard direction, parallel to a crystallographic symmetry axis, can be considered as a second or&r magnetic transition, having its critical point at H = Ha. A macroscopic description of this magnetization process can be given using a phenomenological expression of the anisotropy energy, in terms of the anisotropy constants. The usual expression for uniaxial systems is: where 6 is the angle between the magnetization vector and the c-axis. From a minimization of the free energ? F = Ek HM. sin 0 of an easy axis sy s tem (F' = Ek HM, cos 0 for easy plane), a pheArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19888251 C8 552 JOURNAL DE PHYSIQUE nomenological analytical expression for the magnetization curve and for the anisotropy field along the hard direction Ha = 2 (KI + 2K2 + . -) / M., (H: = 2Kl/ M.) , which is the usually measured &antity, can be'obtained. Fkom these expressions it is evident that in easy-axis systems, the same value of Ha can be obtained with different sets of K's, at a constant M, value; that is the same value of Ha = 2 (Ki + 2K2 + .) / Ma, can correspond to different anisotropy energies Ek = Kl + K2 + . Inversely it should be stressed that systems having the same anisotropy energy (and also the same Ms) can require different field values to be saturated (Ha) . This point is interesting for the possible implications on the coercive force which can indeed depend on the composition of the anisotropy (K's) for the same value of the anisotropy field. A simple analysis can be made in order to quantify the effects of the different anisotropy constants on Hc. The second (Kl) and fourth order (Kz) anisotropy constants will be considered. In presence of domain walls (DW), regardless of the coercivity mechanism, both DW energy (w) and thickness (6) , which are relevant quantities for the determination of the coercive force, depend on all anisotropy constants. The ex~ressions derived for w and 6. considering only K1 [21], are, respectively w = 4 m and 6 = T J ~ , where A is the exchange energy per unit le&h. Using the same method as in reference [21] but introducing a higher order anisotropy constant K2, the following expressions are obtained: W = S ~ G ( T ) and s = T J A / ( K ~ + K ~ ) (2) where r = K 2 / K i a n d ~ ( r ) = l + ( l + r ) ( t a n ' f i ) / f i ) for r > O.(K2 > 0) and for r < 0 (K2 < 0). For r = 0, the previous expressions for w and 6 are obtained. It is interesting to consider the behaviour of w and 6 for constant values of Ha and M. : Ha = 2(1+ 2r) K l / M., thus Kl = HaMs / 2 (1 + 2r), hence: where F (r) = G (r) / d m . The evaluation of these expressions indicate that, for a fixed value of Ha, the presence of a positive K2 anisotropy constant gives rise to a decrease of the DW energy (see Fig. 1) and to an increase of the thickness, whilst the opposite is true for negative K2. In other words an increase of Ha, due to K2 has not the same effect on w and 6 and thus on Hc, as was due to K1. Fig. 1. Dependence on K2 / Kl of: (e) the ratio Hc / Ha for a specimen made of single domain particles, when the coercivity is due to the coherent rotation reversal mech* nism and ( m ) the variation F (T ) of the domain wall energy, where T = K2 / K1, at constant values of Ha and Ms. Also in the case of single domain particles, when a coherent magnetization reversal mechanism determines the coercivity, a dependence of Hc on the anisotropy composition is found. Using the data of reference [22], the dependence of Hc / Ha on the ratio K2 / Kl has been calculated (Fig. I), with a resulting decrease in Hc / Ha with increasing K2 / Kl for positive K2 and an increase for negative K2. It can be concluded that, regardless the coercivity mechanism, Hc depends on the anisotropy composition. This can influence the relative temperature behaviour of Hc and Ha. It can also be observed that F (r) , that is the DW energy variation, shows the same dependence on K2/K1 as Hc/Ha for single domain particle specimen. The curve in figure 1 shows that values of Hc>Ha are allowed. It should be underlined that this refer to a metastable situation of the system, around K2 = K1/2, where Ha = 0 whilst Ek between the two extrema is different from zero, justifying a coercivity. Thus Hc/Ha tends to diverge. Small angle canting model An important aspect of the anisotropy in multisublattice systems is that concerning the interplay between intersublattice exchange, sublattice anisotropies, sublattice moments and external field. The classical phenomenological treatment, which assumes rigid collinear magnetic order, is inadequate to describe systems having high anisotropies: under the action of torques due to competing anisotropies and/or torques due to an external field, in case of different non competing anisotropies, a canting angle between the sublattice moments can be induced. In the framework of a phenomenological description, in terms of anisotropy constants, it has been shown that for a two sublattice system, the occurrence of such a canting angle (small angle), affects the macroscopic anisotropy by a significant amount. However the resultant anisotropy of the system can be still described using effective anisotropy constants, which differ from the simple addition of individual contributions [23-271. The small angle canting model has been shown to have a general validity [25, 261, regardless of the fact that the anisotropies involved are equiverse or opposite and of the ferro or ferrimagnetic coupling between sublattices; the only limitation being that of small angles between individual sublattices and resultant magnetizations. On the other hand this condition is usually verified close to the extrema of the anisotropy energy, that is along the crystallographic symmetry directions 0 = 0 and 0 = 90 [23, 261. This implies that measurable quantities like initial susceptibility (stiffness) and anisotropy field, which can be expressed in terms of anisotropy constants are always good quantities to be used in the framework of the small angle canting (SAC) model. Within such a model, using ad hoc calculations, some contradictions regarding experimental data [24] and some peculiar magnetic behaviours [23, 25, 29, 301 have been explained. The analysis of the proposed models [23,24] suggests two considerations. First, the expressions used to evaluate the effective anisotropy constants can be utilized backward; in the sense that the knowledge of both the effectives anisotropy constants of the system and the real constant of one of the two sublattices, allows the true constants of the second sublattice to be extracted. It is worth noting that the metal sublattice anisotropy is easy to obtain for REintermetallics by measuring the anisotropy in Y or non-magnetic REcompounds. From reference [23] it can be deduced that KI,= ~ l + & + k 3 ~ i b (4) where KI, &, &s are the effective anisotropy constants which describe the resultant system, whilst Kla and Klb are the true intrinsic second order anisotropy constants Kl of the "an and "b" sublattices respectively. As a further issue this simple expression states that the anisotropy field in the hard direction, in the presence of canting, Ha = ( 2 ~ 1 + 4 ~ 2 + . . .) / M,, can be very different from Ha = 2 (Kla + Klb) / Ms without canting, even if the initial (Ek = 0) and final energy states ( ~ k = KI + I& + . . = Kla + ~ i b ) are the same for both situations. Similarly, from reference [24], the true quantity K = Kl + 2K2 (which is the initial susceptibility or stiffness for an easy-plane system; K = KI for easy-axis) of the "a" sublattice can be obtained from the knowledge of both the effective and "b" sublattice K value. A second consideration is that the formation of a caating angle, as a consequence of a relaxation which enable the system to gain energy at the expense of the exchange, results in a hardening of the anisotropy of the system, in the sense that the energy (or field), required to saturate the sample is increased. This is a direct consequence of the fact that the canting has no effects on Ek at 0 = 0 and 0 = 90, as deduced from equation (4), whilst a relaxation reduces the energy of the absolute minimum. On the contrary the stiffness is reduced (softening) as a consequence of canting [23, 241. First order field induced transitions Discontinuous jumps in the magnetization curve along a hard direction, at a critical field value Hcp, have been observed in many systems [5, 6, 11-15, 301. They have been phenomenologically interpreted as first order magnetization processes (FOMP), that is, irreversible rotations of the magnetization vector between two inequivalent states. In this description the conditions for a FOMP are given by the magnetic field, which modifies the free energy surface for a tetragonal system), making a second inequivalent state having the same energy as the absolute minimum (E (01) = E (02)) . Besides H,,, a FOMP is characterized by the magnetization values of the initial Ml and final M2 states. It is easy to show that phenomenological high order anisotropy constants are necessary in order to describe a FOMP [4]. In particular at least 4th (K2) and 6th (K3) order anisotropy constants are required for type I (M2 = M,) and type I1 (M2 < M,) FOMP respectively. It should be underlined that the presence of a type I1 FOMP in easy-axis systems, always requires the existence of a positive K3. Type I and I1 FOMP have been observed respectively in NdaFelrB [5, 61, (with H applied along the [l 0 01) and in PraFelrB [31], in both directions in the basal plane. A phenomenological analysis gives clear evidence of the necessity of the 6th order anisotropy term to describe both systems at low temperature. However for Nd K2 < 0 and K3 < 0 are found whilst the signs are inverted for Pr. This makes the two systems very different, at least in their description. Such different macroscopic behaviour could be related with the results of a recent experiment [32] which gives an indication that the two different crystallographic sites of the rare-earth in the tetragonal2:14:1 structure give the same contribution in Pr2FelrB whilst the contribution of the 4f site is much larger than that of the 4g site in Nd2FelrB. Non-axial anisotropy terms have recently had to be considered in E (0,cp) , in order to justify FOMPs observed when the magnetic field is applied in directions which imply a non co-planarity of M,, c-axis and applied field during the magnetization process, as in Y C8 554 JOURNAL DE PHYSIQUE trigonal ferrites [15] and in Nd2Fe14B [6]. This imposes conditions for the existence of CEF terms BT with m # 0. A combination of a macroscopical technique (SPD) and a phenomenological description allows not only for the measurement of critical parameters but also for the study of FOMP in polycrystals. This can be obtained analyszing the d 2 ~ / d ~ ~ which, according to the SPD theory, has a peak in correspondence of the anisotropy or critical field of a FOMP. The peak shape does not depend on any sample texture because it is due to the crystallites at right angles with.respect to field [16]. Such an analysis has been performed on Pr2 ( F ~ C O ) ~ ~ B (see Fig. 2), SmFellTi and ErFeloV2. In ErFeloV2 compound, which displays a type I FOMP below 150 K, it has been found evidence for a planar contribution of Er to the anisotropy, which is opposite to that expected on the basis of the sign of the second order Stevens coefficent (aJ> 0). This indicate that even at room temperature the sign of the anisotropy cannot be attributable only to aJ [33]. On the other hand the type I1 FOMP observed in SmFenTi [34] is macroscopically (Kz negative and K3 positive) very similar to that observed in Pr2FelrB. A comparative analysis of the systems studied by the author, which show a FOMP when the field is applied in the basal plane, put in evidence that a type I process is observed in intermetallic compounds such as NdzFe14 B, Nd2 (FeCo),, B, PrCos, ErFeloV2 as well as hexaferrites Ba (CoZn), Fe16027 [14]. In all these systems a positive K2 and a negative K3 have found to be necessary in order to describe the process whilst a type I1 FOMP is found in PrzFelsB, Prz (FeCo),, B, SmFellTi where, on the contrary, a negative K2 and a positive K3 are required. Two facts deserve some attention: 1) in all cases K3 has opposite sign with respect to the relative K2 (this is not required by general conditions [4]); 2) K3 is negative in the cases where a competition between different anisotropy contributions exists (type I FOMP). The presence of such a competition is witnessed by the presence of a SRT observed at low temperature in all the systems showing a type I FOMP which is however observed always before that SRT takes place. These two points support or alternatively can be explained in the framework of the small angle canting model which, in case of a two sublattice system predicts the occurrence of a positive K3 if the two sublattices have the same kind of anisotropy whilst K3 can be negative for competing contributions. Furthermore the correcting term has opposite sign in K2 and K3 (Eqs. (6), Ref. [23]). Different types of processes, associated with a transition from a collinear to a non-collinear spin arrangeFig. 2. Measured SPD peakshape at 220 K (a) and 78 K (b) for Prz (Feo.aC0o.4)~~ B. A computer simulation with selected values of the ratios of the anisotropy constants Kz/Ki and K3 / Ki is displayed in (c) and (d). The computer simulation consists of the expected shape of the second derivative (M) with respect to the applied field for a polycrystal. It can be seen that the given parameters reproduce very well the measured SPD peakshape. Additionally, the expected magnetization curve for both poly and single crystal is also displayed. The FOMP is of type 11. (From Ref. 1391.) ment, have been recently observed in hexagonal easy plane ferrimagnetic systems when the field is applied along the t or 6 direction in the basal plane [35-371. Such a transition has been explained in terms of competition between magnetostatic and exchange energies (FOMR). Basal plane anisotropy is not involved because both the two sublattice magnetization vectors, even if canted, always point along two of the six easy basal plane directions [37]. A phenomenologycal description of such transitions, using a two sublattice model, provided satisfactory values for magnetic parameters of the 2:17 compounds. In particular the determination of the 3d-4f exchange parameter has been possible [35-371. A more complex situation has been found in the rhombohedral ReFe3 compounds [lo, 381 where both FOMP and FOMR seem to be simultaneously present. On the occurrence of FOMP in uniaxial systems the following conclusions can be drawn: 1) FOMP are commonly observed in multisublattice systems where multiple contributions to both anisotropy and exchange exist; 2) with the use of a suitable technique (SPD) both the measurement of critical parameters and the analysis of the FOMP can be performed with high accuracy also in polycrystals. The consistency of the analysis is enhanced by general considerations on the type of FOMP; 3) high order anisotropy constants are required in order to describe FOMP. In case of non coplanar magnetization processes non axial anisotmpy terms have to be considered; 4) with reference to systems which are easy-axis at room temperature, type I FOMP (Mz = M.) has been up to now observed in systems where competititions between anisotropy contributions are present. In particular compounds where a easy-axis to easy-cone transition takes place at low temperatures. Regardless of the type of compound, the description of such a process has always required a negative 6th order and a positive 4th order anisotropy constants; 5) type I1 FOMP has been observed in compounds where different contributions to the anisotropy seem to have the same sign. In these cases a positive K3 and a negative Kz are required to describe the observed process; 6) there is consistence between the occurrence of FOMP (type, necessity of high order constants, sign of constants) and the small angle canting (SAC) model. It should be underlined that the SAC model has furnished interesting indications and should deserve some more attention in the future. [I] Sagawa, M., Fujimura, S., Togawa, M. and Matsuura, Y., J. Appl. Phys. 55 (1984) 29. [2] Buschow, K. H. J., Handbook on ferromagnetic materials, vol. 2, Ed. E. P. Wohlfarth (NorthHolland, Amsterdam) 1980, p. 297. [3] Smit, J. and Wijn, H. P., Ferrites (Philips Techn. Lib., Eindhoven) 1959. [4] Asti, G. and Bolzoni, F., J. Magn. Magn. Mater. 20 (1980) 29. [5] Pareti, L., Boboni, F. and Moze, O., Phys. Rev. B 32 (1985) 7604. [6] Bolzoni, F., Moze, 0. and Pareti, L., J. Appl. Phys. 62 (1987) 615. [7] Yamada, M., Yamaguchi, Y., Kato, H., Yamamoto, H., Nakagawa, y., Hirosawa, S. and Sagawa, M., Solid State Commun. 56 (1985) 663. [8] Hirosawa, S. and Sagawa, M., Solid State Commun. 54 (1985) 335. [9] Kebe, B., James, W. J., Deportes, J., Lemaire, R., Yelon, W. and Day, R. K., J. Appl. Phys. 52 (1981) 2052. [lo] Jin, L., James, W. J., Lemaire, R. and Rhyne, J., J. Less Common. Met. 118 (1986) 269. [ l l ] Ermolenko, A. S. and Rozda, A. F., IEEE Trans. Magn. MAG-14 (1978) 676. [12] Lee R. W., IEEE Trans. Magn. MAG-15 (1979) 762. [13] Asti, G., Bolzoni, F., Licci, F. and Canali, M., IEEE Trans. Magn. MAG-14 (1978) 883. [14] Paoluzi, A., Licci, F., Moze, O., Turilli, G., Deriu, A., Albanese, G. and Calabrese, E., J. Appl. Phys. 63 (1988) 5074. [15] Bolzoni, F. and Pareti, L., J. Magn. Magn. Mater. 42 (1984) 44. [16] Asti, G. and Rinaldi, S., J. Appl. Phys. 45 (1974) 3600. [17] Pareti, L., Solzi, M., Bolzoni, F., Moze, 0. and Panizzieri, R., Solid State Cornmun. 61 (1987) 761. [18] Bolzoni, F., Coey, J. M. D., Gavigan, J., Givord, D., Moze, O., Pareti, L. and Viadieu, T., J. Magn. Magn. Mater. 65 (1987) 123. [I91 Grossinger, R., Krewenka, R., Sun, X. K., Buschow, K. H. J., Eibler, R. and Kirchmayr, H. R., J. Less Common. Met. 124 (1986) 165. [20] Bolzoni, F., Leccabue, F., Pareti, L. and Sanchez, J. L., J. Phys. Cofloq. France 46 (1985) C6-305. (211 Tebble, R. S., Magnetic domains (Methuen and . . Acknowledgment CO., London) 1969' p. 34. [22] Gerling, W., Zeitschript fur angewandte Physik Valuable discussions with prof. G. Asti, dr. 0. 28 (1969) 4. Moze, dr. A. Paoluzi and dr. G. Turilli are greatly [23] Rinaldi, S. and Pareti, .L., J. Appl. Phys. 50 appreciated. (1979) 779. C8 556 JOURNAL DE PHYSIQUE 1241 Sarkis, A. and Callen, E., Phys. Rev. B 26 (1982) 3870. [25] Asti, G. and Deriu, A., Proc. of the VI Int. Workshop on Re-Co Permanent Magnets and Their Applications and I11 Int. Symp. on Magnetic Anisotropy and Coercivity in RETM Alloys Ed. J. Fidler, Tech. Univ. of Vienna (1982) p. 525. 1261 Asti, G., IEEE Trans. Magn. MAG-17 (1988) 2630. [27] Radwanski, R. J., Franse, J. J. M. and Sinnema, S., J. Magn. Magn. Mater. 70 (1987) 33. [28] Ermolenko, A. S., IEEE Trans. Magn. MAG-15 (1979) 1765. [29] Asti, G., Bolzoni, F., Leccabue, F., Panizzieri, R., Pareti, L. and Rinaldi, S., J. Magn. Magn. Mater. 15-18 (1980) 561. [30] Pareti, L., Moze, O., Solzi, M. and Bolzoni, F., J. Appl. Phys. 63 (1988) 172. [31] Hiroyoshi, H., Kato, H., Yamada, M., Saito, N., Nakagawa, Y., Hirosawa, S., Sagawa, M., Solid State Commm. 62 (1987) 475. [32] Moze, O., Pareti, L., Marusi, G. and David, W. I. F., Int. Conf. Neutron Scattering, Grenoble (July 12-15, 1988). 1331 Solzi, M., Pareti, L., Mme, O., ICM-Paris (July 25-29, 1988). [34] Hong Huo, Li, Boping, Hu, Gavigan, J. P., Coey, J. M. D., Pareti, L. and Moze, O., ICM-Paris (July 25-29, 1988). [35] Franse, J. J. M., de Boer, F. R., Rings, P. H., Gersdorf, R., Menovsky, A., Muller, F. A., Radwanski, R. J. and Sinnema, S., Phys. Rev. B 31 (985) 4347. 1361 Sinnema, S., Franse, J. J. M., Radwanski, R. J., Menovski, A. and de Boer, F. R., J. Phys. F. 17 (1987) 233. [37} Radwanski, R. J., Franse, J. J. M. and Sinnema, S., J. Magn. Magn. Mater. 51 (1985) 175. 1381 Jin, L., James, W. J., Lemaire, R. and Rhyne, J., J. Less. Common Met. 118 (1986) 269. [39] Pareti, L., Moze, O., Solzi, M., Bolzoni, F., Asti, G. and Marusi, G., CEAM Report, Madrid (1988).