Random anisotropy effects in soft magnetic nanocrystalline materials

ARTICLE IN PRESS

Physica B 372 (2006) 256–259 www.elsevier.com/locate/physb

Random anisotropy effects in soft magnetic nanocrystalline materials J.G. Labiano, C. Go´mez-Polo Departamento de Fı´sica, Universidad Pu´blica de Navarra, Campus de Arrosadı´a, 31006 Pamplona, Spain

Abstract The basic magnetic parameters (coercivity, remanent magnetization) of the magnetization process in nanocrystalline soft magnetic alloys are analysed. The estimation procedure mainly consists of the energy minimization of an assembly of single domain particles, taking into account the main contributions in these biphasic systems (crystalline and residual amorphous phases). The grain size dependence of coercivity in these nanocrystalline materials is analysed considering the occurrence of long-range exchange interactions exponentially attenuated as a function of the distance between interacting particles. r 2005 Elsevier B.V. All rights reserved. PACS: 75.50.Tt; 75.10.Hk; 75.30.Gw Keywords: Magnetization process; Nanocrystalline materials; Grain size dependence

1. Introduction The magnetic behaviour of soft magnetic nanocrystalline materials has been extensively studied during the last decade [1]. The interest in these materials is mainly related with the search of new bulk Fe-based soft magnetic alloys for inductive applications, being now commercially available under different trade marks (i.e. FINEMETTM, Hitachi-Metals Co. or VITROPERM, Vacuumschmelze GmbH). However, the magnetic properties of these heterogeneous magnetic systems represent a research topic of continuous interest from both experimental and theoretical points of view. The macroscopic soft magnetic behaviour of these nanocrystalline materials is a direct consequence of their ultrafine grain structure composed of randomly oriented BCC Fe-rich crystals embedded in a residual amorphous matrix. Their excellent soft magnetic response arises from the averaging out of the magnetocrystalline anisotropy via the magnetic interactions between the two constituent magnetic phases. Herzer [1] successfully applied the random anisotropy model to explain the averaging out of the magnetocrystalline anisotropy in these biphasic sysCorresponding author. Tel.: +34 948 169576; fax: +34 948 169565.

E-mail address: [email protected] (C. Go´mez-Polo). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.061

tems. This model was initially developed by Alben et al. [2] to explain the soft magnetic behaviour of amorphous ferromagnets. The relevant parameter in this theory is the exchange correlation length, L0, over which the local magnetization changes. Assuming the orientation fluctuation length of randomly distributed local easy axis equal to the average crystalline size, d, the magnetocrystalline anisotropy, K, is averaged out over several grains when the condition L0 bd is fulfilled. Under these conditions, the effective anisotropy constant hKi is averaged out over N grains randomly oriented within L0 ðN ¼ ðL0 =dÞ3 Þ:  3=2 K d hKi ¼ pffiffiffiffiffi ¼ K . (1) L0 N Since L0 represents the critical scale where the exchange energy starts to balance the anisotropy energy, it can be directlypffiffiffiffiffiffiffiffiffiffiffiffiffi associated with the domain wall thickness ffi (L0 ¼ A=hKi; A exchange constant). The above statistical and scaling arguments have been extensively employed to explain the main experimental features of these nanocrystalline systems [3,4]. However, this model is strictly valid for a single phase crystalline material. In order to take into account the more realistic multiphase nature of these systems, different modified models have been proposed considering the role of the residual amorphous matrix as exchange transmitter [5,6].

ARTICLE IN PRESS J.G. Labiano, C. Go´mez-Polo / Physica B 372 (2006) 256–259

2. Estimation procedure Starting from the original work of Alben et al. [2], a random anisotropy ferromagnet can be described through the simplest model Hamiltonian representing a random uniaxial anisotropy in an exchange coupled spin system by  X X  ~j
~i Þ2
1 ~i  S H¼
J ij S Di ðn^ i  S 3 iaj i X ~i , ~ ext 
H gmB S ð2Þ

particles (z): J ¼ J 0 expð
azÞ

with a a characteristic parameter of the exchange coupled system. The minimum energy configuration of the system was obtained through the energy minimization of each individual particle, where the final spin configuration was achieved after successive iterations. Starting from a random distribution, the magnetization of the particle was forced to align in the opposite direction with respect to each Hamiltonian gradient. The hysteresis loops and the associated characteristic parameters (coercivity, HC, and remanent magnetization, Mr) were obtained starting from the saturated state (i.e. all the particles aligned parallel to ~ext ). The energy minimization procedure was successively H repeated starting from different random distributions and the mean values of HC and Mr were obtained. The performed iteration procedure guarantees a standard deviation of the displayed parameters close to 1%. 3. Results and discussion Fig. 1 shows the reduced values of the coercive field (hC ¼ H C =H K ; H K ¼ 2K=m0 M S ) and magnetization at the remanence (mr ¼ M r =M S ) for a monophasic system composed of randomly oriented uniaxial single domain particles of BCC Fe–Si (20 at%) (K ¼ 15  103 J=m3 , J 0 ¼ 2  10
21 J, a ¼ 2  106 m, M S ¼ 1:75  106 A=m, R ¼ 350 nm). For high grain sizes, the displayed results correspond with those obtained in a set of non-interacting uniaxial single domain particles (hC  0:5; mr  0:5). In this region, the coercivity fits to the well-known 1/d experimental law. However, the reduction of d below a critical value (dc), gives rise, as Fig. 1 shows, to the characteristic averaging out of the magnetocrystalline anisotropy in a random anisotropy system and thus to a remarkable reduction of the coercivity of the system. Therefore, dc can be directly correlated with the 0.8

i

1.0 Reduced coercive field, hc

~i is the spin at site i, Jij the exchange coupling where S constant between spins i and j, Di and n^ i the local uniaxial anisotropy and a unit vector along the easy axis direction ~ ext the external magnetic field and at site i, respectively, H ~i the magnetic moment at site i. In our model, the gmB S ~i are substituted by uniaxial spherical individual spins S ferromagnetic particles randomly distributed within a spherical volume of radius R. Thus, the characteristic anisotropy constant K and saturation magnetization Ms are expressed in terms of the mean grain diameter of the spherical particles, d: D ¼ KV and gmB ¼ m0 M S V , with V ¼ 4=3pðd=2Þ3 . Moreover, the exchange coupling constant between adjacent ferromagnetic particles is introduced as a spatial dependent function, exponentially attenuated with the distance between the interacting

(3)

0.6 0.8 0.4 0.6

Reduced remanence, mr

Thus, some characteristic experimental facts (magnetic hardening at the initial stages of nanocrystallization and close to the paramagnetic transition of the residual amorphous phase) can be understood in terms of effect of the residual amorphous matrix contribution [7–9]. The aim of the work is to present a different approach to the analysis of the magnetization process and basic magnetic parameters (coercivity, remanent magnetization) in these nanocrystalline alloys. The estimation procedure mainly consists of the energy minimization of an assembly of single domain particles, taking into account the main contributions in these systems: magnetocrystalline anisotropy, magnetic energy associated with the applied field and exchange anisotropy. The estimation constitutes the extension of simplest Hamiltonian model representing random uniaxial anisotropy in an exchange coupled spin system [2], where each individual spin is substituted by a ferromagnetic particle with mean grain size d and magnetic moment m ¼ M S V (MS saturation magnetization, V ¼ 4=3pðd=2Þ3 ). The grain size dependence of the coercivity of these systems is analysed taking into account the occurrence of long-range exchange interactions exponentially attenuated as a function of the distance between the ferromagnetic particles. The developed model basically reproduces the main experimental facts in these nanocrystalline systems, i.e., the occurrence of a exchange coupled state for mean grain sizes below a critical value and the effect of the residual amorphous matrix in the exchange coupling of the ferromagnetic grains.

257

0.2 25

30

35 d (nm)

40

45

Fig. 1. Reduced coercive field (K), hC ¼ H C =H K , and reduced remanence (J) mr ¼ M r =M S , as a function of the grain diameter, d, for a single phase system (K ¼ 15  103 J=m3 , J 0 ¼ 2  10
21 J, a ¼ 2  106 m, M S ¼ 1:75 106 A=m, R ¼ 350 nm).

ARTICLE IN PRESS J.G. Labiano, C. Go´mez-Polo / Physica B 372 (2006) 256–259

characteristic pffiffiffiffiffiffiffiffiffiffi ferromagnetic exchange correlation length ðL0 ¼ A=K Þ, where the exchange energy starts to balance the anisotropy term. Moreover, as it should be expected from the assumed distance-dependent exchange coupling constant, the estimated curves depend on the particular value of a: an increase in a gives rise to a decrease in the effective exchange correlation length and thus to a parallel decrease in the critical grain diameter, dc. Thus, the selected value of a in Fig. 1 corresponds with a critical grain diameter dc equal to the estimated value of the exchange correlated length in BCC Fe–Si (20 at%) (L0  30 nm). In order to adapt this model to biphasic systems, the exchange interaction between both ferromagnetic phases is described in the following terms: J ij ¼ J 0ij expð
azÞ, pffiffiffiffiffiffiffiffiffiffiffiffi J 0ij ¼ J 0i J 0j ,

ð4Þ

where the subscripts i ¼ 1 and j ¼ 2 would apply in the case of soft nanocrystalline samples to the crystalline (high anisotropy) and soft magnetic amorphous phases, respectively. Fig. 2 displays the dependence of hC and mr on the grain diameter for a two phase system composed by 80% of BCC Fe–Si (20 at%) crystalline phase (subscript 1). In order to simulate the effect of the residual amorphous phase, the anisotropy constant of the remaining 20% ferromagnetic particles (subscript 2) has been reduced 2 orders of magnitude (K 1 =K 2 ¼ 102 ). As observed, while the reduced remanence does not significantly change, the main effect of the introduction of the low anisotropy phase is to promote an overall reduction in the coercive field of the coupled system. This simple biphasic model can also be employed to outline the main characteristics of the evolution of the coercivity with the volume crystalline fraction in soft magnetic nanocrystalline alloys. In particular, the occurrence of a magnetic hardening (increase of the sample

coercivity) under those circumstances where the soft magnetic phase loses its exchange transmission capacity. Fig. 3 displays the estimated coercive field as a function of the volume percentage, n1, of phase 1 (BCC Fe–Si (20 at%) nanocrystals) for mean grain diameter d ¼ 22 nm (K 1 ¼ 15  103 J=m3 , K 2 ¼ 15 J=m3 , J 01 ¼ 2  10
21 J, J 02 ¼ 2  10
26 J, a ¼ 2  106 m, MS1 ¼ MS2 ¼ 1.75  106 A/m, R ¼ 220 nm). For n1  1 (fully crystallized samples), the exchange coupling between adjacent crystallites promotes an average out of the magnetocrystalline anisotropy (hC  0:2). A diminution in n1 promotes a smooth decrease of the coercivity of the system as a consequence of the reduction, with the inclusion of the soft magnetic phase, of the mean magnetocrystalline anisotropy. For n1 p0:5, the coercivity drastically decreases indicating the main influence of the soft magnetic phase on the magnetization process of the system. It is remarkable that this abrupt magnetic softening is only detected for biphasic systems with dissimilar magnetocrystalline anisotropy values (K 1 4100 K2). However, for 0:55pn1 p0:75, the increase in the percentage of the soft magnetic phase has associated an opposite trend, i.e., the occurrence of a slight magnetic hardening (a maximum value in HC, see Fig. 3). The appearance of this maximum should be understood in terms of a partial magnetic decoupling of the high anisotropy ferromagnetic crystallites (phase 1). Thus, in this range of n1, as a consequence of the lower exchange correlation length of the low anisotropy phase (J 01 ¼ 2  10
21 J and J 02 ¼ 2  10
26 J), this soft magnetic phase in not able to effectively exchange couple the ferromagnetic grains of phase 1 with higher magnetocrystalline anisotropy. For volume fractions of the soft magnetic phase above a critical value (n1 o0:5 in this estimation), the reduction of the effective magnetocrystalline anisotropy dominates the overall behaviour, giving rise to the detected sharp decrease in the coercive field of the coupled system.

0.6 0.8 0.4 0.6

Reduced remanence, mr

Reduced coercive field, hc

1.0

0.2

Reduced coercive field, hc

0.8

1.0 0.2 0.8 0.1 0.6

0.0

0.4 0.0

25

30

35 d (nm)

40

45

Fig. 2. Reduced coercive field (K), hC ¼ HC/HK1, and reduced remanence (J) mr ¼ Mr/MS1, as a function of the grain diameter, d, for a two phase system with 80% of crystalline phase (K 1 ¼ 15  103 J=m3 , K 2 ¼ 150 J=m3 , J 01 ¼ J 02 ¼ 2  10
21 J, a ¼ 2  106 m, M S1 ¼ M S2 ¼ 1:75  106 A=m, R ¼ 350 nm).

Reduced remanence, mr

258

0.2

0.4 0.6 0.8 Volume fraction ν1

1.0

Fig. 3. Reduced coercive field (K), hC ¼ HC/HK1, and reduced remanence (J) mr ¼ Mr/MS1 as a function of the volume percentage, n1, of the higher anisotropy phase (BCC Fe–Si (20 at%) nanocrystals) for mean grain diameter d ¼ 22 nm (K 1 ¼ 15  103 J=m3 , K 2 ¼ 15 J=m3 , J 01 ¼ 2 10
21 J, J 02 ¼ 2  10
26 J, a ¼ 2  106 m, MS1 ¼ MS2 ¼ 1.75  106 A/m, R ¼ 220 nm).

ARTICLE IN PRESS J.G. Labiano, C. Go´mez-Polo / Physica B 372 (2006) 256–259

259

4. Conclusions

Acknowledgement

To summarize, the magnetization process in random anisotropy nanocrystalline systems is analysed in terms of the energy minimization of an assembly of single domain particles. The exchange interaction between adjacent ferromagnetic particles is introduced as an exponentially attenuated function with the distance between the interacting particles. The proposed model is able to reproduce the characteristic averaging out of the magnetocrystalline anisotropy in exchange coupled systems with easy axis randomly oriented for structural correlation lengths below a critical size, dc. The estimated procedure is extended to a biphasic system composed of two ferromagnetic phases with different magnetocrystalline anisotropy constants. Within the framework of this simple biphasic model, the main experimental features of soft nanocrystalline alloys are suitably reproduced. In particular, the enhancement of the soft magnetic response in this exchange coupled systems with the inclusion of a low anisotropy soft magnetic phase and the occurrence of a magnetic hardening when the soft magnetic phase decreases its exchange transmission capacity.

The authors would like to thank Dr. B. Bujanda (Dept. De Matematica Aplicada, Universidad Pu´blica de Navarra) for helpful discussions during the elaboration of this work. References [1] G. Herzer, Nanocrystalline soft magnetic alloys, in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials, vol. 10, Elsevier, Amsterdam, 1997, p. 415. [2] R. Alben, J.J. Becker, M.C. Chi, J. Appl. Phys. 49 (3) (1978) 1653. [3] A. Hernando, T. Kulik, Phys. Rev. B 49 (1994) 7064. [4] K. Hono, M. Ohnuma, in: H.S. Nalwa (Ed.), Magnetic Nanostructures, American Scientific Publishers, Stevenson Ranch, California, 2002, p. 327 (Chapter 8). [5] A. Hernando, M. Va´zquez, T. Kulik, C. Prados, Phys. Rev. B 51 (1995) 3581. [6] K. Suzuki, J.M. Cadogan, Phys. Rev. B 58 (1998) 2730. [7] C. Go´mez-Polo, D. Holzer, M. Multigner, E. Navarro, P. Agudo, A. Hernando, M. Va´zquez, H. Sassik, R. Gro¨ssinger, Phys. Rev. B 53 (1996) 3392. [8] J. Arcas, A. Hernando, C. Go´mez-Polo, F.J. Castan˜o, M. Va´zquez, A. Neuweiler, H. Kro¨nmuller, J. Phys.: Condens. Matter 12 (2000) 3255. [9] C. Go´mez-Polo, P. Marı´ n, L. Pascual, A. Hernando, M. Va´zquez, Phys. Rev. B 65 (2002) 24433.