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Hysteresis and thermal relaxation in nanocrystalline soft magnetic materials
Journal of Magnetism and Magnetic Materials 215}216 (2000) 446}448
Hysteresis and thermal relaxation in nanocrystalline soft magnetic materials M. LoBue*, V. Basso, P. Tiberto, C. Beatrice, G. Bertotti Physics Department, IEN Galileo Ferraris and INFM, Corso Massimo d+Azeglio 42, I-10125 Torino, Italy
Abstract Hysteresis and relaxation properties are studied in nanocrystalline Finemet-type materials beyond the Curie point of the amorphous matrix. Measurements of hysteresis loops and return branches at a temperature near the maximum hardening are used to investigate the microstructure of the material. Data are discussed by a uni"ed model of hysteresis and thermal activation. 2000 Elsevier Science B.V. All rights reserved. Keywords: Magnetic after-e!ect; Hysteresis modeling; Thermal activation
Thermally activated phenomena play a relevant role in hard magnetic materials. Traditionally, substantial interest has been devoted to this subject in the "eld of permanent magnet applications and data storage. The presence of after-e!ect phenomena in soft magnetic materials is generally small and di$cult to detect. However, it can become relevant in nanocrystalline materials, where the fact that the activation volumes are small, makes thermal activation more important [1]. In this paper, we study the properties of magnetization curves and thermal relaxation in nanocrystalline Fe Cu Nb Si B (Finemet) alloys [2] consisting of Fe}Si crystallites embedded in an amorphous matrix. The two phases are ferromagnetic at room temperature, where Finemet presents outstanding soft magnetic properties. However, they have very di!erent Curie temperatures (¹&3503C for the amorphous matrix and ¹&6003C for the nanocrystalline phase). When the material is brought beyond the Curie temperature of the amorphous matrix, the magnetic decoupling of nano-grains generates a strong coercivity increase known as magnetic hardening [3], a drastic decrease of the activation volumes and consequently a dramatic enhancement of relaxation phenomena. We investigated the magnetic properties of two singlestrip samples (30 cm long, 10 mm wide, 20 lm thick) * Corresponding author. Tel.: #39-011-39-19753; fax: #39011-650-7611. E-mail address: [email protected] (M. LoBue).
prepared by rapid solidi"cation and annealed in a furnace at 5503C for 1 h (sample A) and at 5003C for 1 h and 30 min (sample B). The volume fraction occupied by Fe}Si crystallites is estimated to be 70% for sample A and 30% for sample B [4], whereas the average grain size is expected to be around 10 nm for both samples. We measured: (1) hysteresis loops under controlled triangular "eld waveform ("eld rate dH/dt in the range 10}10 Am\s\), with particular attention to the dependence of the coercive "eld H on the "eld rate dH/dt; (2) return branches along the major hysteresis loop; (3) time decay of the magnetization M(t;H , dH/dt) at di!er ent "elds H reached under di!erent "eld rates dH/dt. All measurements were performed at a temperature around the maximum of the hardening e!ect (¹"4303C for sample A and ¹"4103C for sample B, see Fig. 1). Magnetization frequencies were always low enough to avoid any appreciable eddy-current e!ect. Experiments were interpreted by a uni"ed description of hysteresis and thermally activated phenomena, recently proposed [1,5]. In this approach, the system is assumed to be an assembly of elementary two-level-subsystems distributed in energy levels and energy barriers. The state of the assembly is described by the timedependent function b(h , t), 0)h )R, obeying the equation:
H h H(t)!b(h , t) *b(h , t) "2 sinh exp ! , q H H *t
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 1 8 5 - 2
(1)
M. LoBue et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 446}448
Fig. 1. Coercive "eld H versus temperature. Open circles: sample A (annealed at 5503C), dH/dt"2;10 Am\ s\. Solid squares: sample B (annealed at 5003C), dH/dt" 5.5;10 Am\ s\. The arrows indicate temperatures at which data of Figs. 2}5 were measured.
447
Fig. 2. Hysteresis loop and return branches measured on sample A at ¹"4303C, dH/dt"2;10 Am\ s\.
where H(t) is the magnetic "eld, q 10\ s is a typical attempt time and H is the #uctuation "eld. One can express H as H "k ¹/k M v, where k is the Bol tzmann constant, ¹ is the absolute temperature, M is the saturation magnetization and v is the activation volume. Past history enters the description through the initial condition b(h , t"0). Once the time-dependent solution b(h , t) is worked out, one calculates the magnetization by the equation: M"2M Q
@F R dh p(h , h ) dh ,
(2)
where p(h , h ) is the so-called Preisach distribution. Eqs. (1) and (2) provide a suitable basis for the interpretation of all the experiments previously mentioned. As a "rst comment, we stress the fact that the experiments beyond the Curie point of the amorphous matrix amplify certain microstructural aspects that are not at all evident at room temperature. In particular, Figs. 2 and 3 give evidence of remarkable di!erences in the hysteresis loop shape of samples A and B. The lower coercive "eld of sample A indicates that larger clusters of coupled nanograins are here involved in the magnetization process. This conclusion is con"rmed by the behavior of H . The #uctuation "eld can be estimated by measuring hysteresis loops traversed at di!erent magnetization
Fig. 3. Hysteresis loop and return branches measured on sample B at ¹"4103C, dH/dt"5.5;10 Am\ s\.
frequencies, and by determining the dependence of the coercive "eld H on the "eld rate dH/dt (Figs. 4 and 5). By solving Eqs. (1) and (2), one predicts H H ln("dH/dt")#C. The insets in Figs. 4 and 5 show that this law is in very good agreement with experiments. One "nds H 8 Am\ for sample A and H 55 Am\ for sample B, which gives an activation volume of linear dimension of the order of 100 and 50 nm, respectively. As
448
M. LoBue et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 446}448
Fig. 4. Hysteresis loops measured on sample A at ¹"4303C under di!erent "eld rates. Inset shows dependence of coercive "eld H on "eld rate dH/dt.
Fig. 5. Hysteresis loops measured on sample B at ¹"4103C under di!erent "eld rates. Inset shows dependence of coercive "eld H on "eld rate dH/dt.
previously anticipated, the activation volume is smaller for the sample of higher coercivity. The interplay of hysteresis and relaxation e!ects clearly emerges when one measures return branches along the major hysteresis loop (see Figs. 2 and 3). One obtains a set of non-monotone curves, which in principle should be predicted by solving Eqs. (1) and (2) for the various relevant "eld histories and by making some suitable assumption for the Preisach distribution p(h , h ). In this respect, a particular aspect easily amenable to an experimental test is the theory prediction for the behavior of the di!erential susceptibility v at a given reversal "eld H . One "nds that, under triangular "eld excitation, the susceptibilities just before and after the turning point are opposite to each other, and the susceptibility after the turning point follows the law v(H)J[2exp ("H!H "/H )!1]\. Remarkably, this law is independent of the "eld rate dH/dt. We veri"ed that all these predictions are in agreement with
experiments if one assumes for H the same value found from the coercive "eld analysis (see Figs. 4 and 5). This is an important con"rmation of the consistency of the entire approach. Additional information can be obtained by studying the magnetization decay M(t;H ,dH/dt) at di!erent "elds H reached under di!erent "eld rates dH/dt. These aspects are discussed in Ref. [5].
References [1] V. Basso, M. LoBue, C. Beatrice, P. Tiberto, G. Bertotti, IEEE Trans. Magn. 34 (1998) 1177. [2] Y. Yoshizawa et al., J. Appl. Phys. 64 (1988) 6044. [3] A. Hernando, T. Kulik, Phys. Rev. B 49 (1994) 7064. [4] P. Allia, M. Baricco, P. Tiberto, F. Vinai, J. Appl. Phys. 74 (1993) 3137. [5] V. Basso, C. Beatrice, M. LoBue, P. Tiberto, and G. Bertotti, Phys. Rev. B 6, (2000) 1278.
Hysteresis and thermal relaxation in nanocrystalline soft magnetic materials M. LoBue*, V. Basso, P. Tiberto, C. Beatrice, G. Bertotti Physics Department, IEN Galileo Ferraris and INFM, Corso Massimo d+Azeglio 42, I-10125 Torino, Italy
Abstract Hysteresis and relaxation properties are studied in nanocrystalline Finemet-type materials beyond the Curie point of the amorphous matrix. Measurements of hysteresis loops and return branches at a temperature near the maximum hardening are used to investigate the microstructure of the material. Data are discussed by a uni"ed model of hysteresis and thermal activation. 2000 Elsevier Science B.V. All rights reserved. Keywords: Magnetic after-e!ect; Hysteresis modeling; Thermal activation
Thermally activated phenomena play a relevant role in hard magnetic materials. Traditionally, substantial interest has been devoted to this subject in the "eld of permanent magnet applications and data storage. The presence of after-e!ect phenomena in soft magnetic materials is generally small and di$cult to detect. However, it can become relevant in nanocrystalline materials, where the fact that the activation volumes are small, makes thermal activation more important [1]. In this paper, we study the properties of magnetization curves and thermal relaxation in nanocrystalline Fe Cu Nb Si B (Finemet) alloys [2] consisting of Fe}Si crystallites embedded in an amorphous matrix. The two phases are ferromagnetic at room temperature, where Finemet presents outstanding soft magnetic properties. However, they have very di!erent Curie temperatures (¹&3503C for the amorphous matrix and ¹&6003C for the nanocrystalline phase). When the material is brought beyond the Curie temperature of the amorphous matrix, the magnetic decoupling of nano-grains generates a strong coercivity increase known as magnetic hardening [3], a drastic decrease of the activation volumes and consequently a dramatic enhancement of relaxation phenomena. We investigated the magnetic properties of two singlestrip samples (30 cm long, 10 mm wide, 20 lm thick) * Corresponding author. Tel.: #39-011-39-19753; fax: #39011-650-7611. E-mail address: [email protected] (M. LoBue).
prepared by rapid solidi"cation and annealed in a furnace at 5503C for 1 h (sample A) and at 5003C for 1 h and 30 min (sample B). The volume fraction occupied by Fe}Si crystallites is estimated to be 70% for sample A and 30% for sample B [4], whereas the average grain size is expected to be around 10 nm for both samples. We measured: (1) hysteresis loops under controlled triangular "eld waveform ("eld rate dH/dt in the range 10}10 Am\s\), with particular attention to the dependence of the coercive "eld H on the "eld rate dH/dt; (2) return branches along the major hysteresis loop; (3) time decay of the magnetization M(t;H , dH/dt) at di!er ent "elds H reached under di!erent "eld rates dH/dt. All measurements were performed at a temperature around the maximum of the hardening e!ect (¹"4303C for sample A and ¹"4103C for sample B, see Fig. 1). Magnetization frequencies were always low enough to avoid any appreciable eddy-current e!ect. Experiments were interpreted by a uni"ed description of hysteresis and thermally activated phenomena, recently proposed [1,5]. In this approach, the system is assumed to be an assembly of elementary two-level-subsystems distributed in energy levels and energy barriers. The state of the assembly is described by the timedependent function b(h , t), 0)h )R, obeying the equation:
H h H(t)!b(h , t) *b(h , t) "2 sinh exp ! , q H H *t
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 1 8 5 - 2
(1)
M. LoBue et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 446}448
Fig. 1. Coercive "eld H versus temperature. Open circles: sample A (annealed at 5503C), dH/dt"2;10 Am\ s\. Solid squares: sample B (annealed at 5003C), dH/dt" 5.5;10 Am\ s\. The arrows indicate temperatures at which data of Figs. 2}5 were measured.
447
Fig. 2. Hysteresis loop and return branches measured on sample A at ¹"4303C, dH/dt"2;10 Am\ s\.
where H(t) is the magnetic "eld, q 10\ s is a typical attempt time and H is the #uctuation "eld. One can express H as H "k ¹/k M v, where k is the Bol tzmann constant, ¹ is the absolute temperature, M is the saturation magnetization and v is the activation volume. Past history enters the description through the initial condition b(h , t"0). Once the time-dependent solution b(h , t) is worked out, one calculates the magnetization by the equation: M"2M Q
@F R dh p(h , h ) dh ,
(2)
where p(h , h ) is the so-called Preisach distribution. Eqs. (1) and (2) provide a suitable basis for the interpretation of all the experiments previously mentioned. As a "rst comment, we stress the fact that the experiments beyond the Curie point of the amorphous matrix amplify certain microstructural aspects that are not at all evident at room temperature. In particular, Figs. 2 and 3 give evidence of remarkable di!erences in the hysteresis loop shape of samples A and B. The lower coercive "eld of sample A indicates that larger clusters of coupled nanograins are here involved in the magnetization process. This conclusion is con"rmed by the behavior of H . The #uctuation "eld can be estimated by measuring hysteresis loops traversed at di!erent magnetization
Fig. 3. Hysteresis loop and return branches measured on sample B at ¹"4103C, dH/dt"5.5;10 Am\ s\.
frequencies, and by determining the dependence of the coercive "eld H on the "eld rate dH/dt (Figs. 4 and 5). By solving Eqs. (1) and (2), one predicts H H ln("dH/dt")#C. The insets in Figs. 4 and 5 show that this law is in very good agreement with experiments. One "nds H 8 Am\ for sample A and H 55 Am\ for sample B, which gives an activation volume of linear dimension of the order of 100 and 50 nm, respectively. As
448
M. LoBue et al. / Journal of Magnetism and Magnetic Materials 215}216 (2000) 446}448
Fig. 4. Hysteresis loops measured on sample A at ¹"4303C under di!erent "eld rates. Inset shows dependence of coercive "eld H on "eld rate dH/dt.
Fig. 5. Hysteresis loops measured on sample B at ¹"4103C under di!erent "eld rates. Inset shows dependence of coercive "eld H on "eld rate dH/dt.
previously anticipated, the activation volume is smaller for the sample of higher coercivity. The interplay of hysteresis and relaxation e!ects clearly emerges when one measures return branches along the major hysteresis loop (see Figs. 2 and 3). One obtains a set of non-monotone curves, which in principle should be predicted by solving Eqs. (1) and (2) for the various relevant "eld histories and by making some suitable assumption for the Preisach distribution p(h , h ). In this respect, a particular aspect easily amenable to an experimental test is the theory prediction for the behavior of the di!erential susceptibility v at a given reversal "eld H . One "nds that, under triangular "eld excitation, the susceptibilities just before and after the turning point are opposite to each other, and the susceptibility after the turning point follows the law v(H)J[2exp ("H!H "/H )!1]\. Remarkably, this law is independent of the "eld rate dH/dt. We veri"ed that all these predictions are in agreement with
experiments if one assumes for H the same value found from the coercive "eld analysis (see Figs. 4 and 5). This is an important con"rmation of the consistency of the entire approach. Additional information can be obtained by studying the magnetization decay M(t;H ,dH/dt) at di!erent "elds H reached under di!erent "eld rates dH/dt. These aspects are discussed in Ref. [5].
References [1] V. Basso, M. LoBue, C. Beatrice, P. Tiberto, G. Bertotti, IEEE Trans. Magn. 34 (1998) 1177. [2] Y. Yoshizawa et al., J. Appl. Phys. 64 (1988) 6044. [3] A. Hernando, T. Kulik, Phys. Rev. B 49 (1994) 7064. [4] P. Allia, M. Baricco, P. Tiberto, F. Vinai, J. Appl. Phys. 74 (1993) 3137. [5] V. Basso, C. Beatrice, M. LoBue, P. Tiberto, and G. Bertotti, Phys. Rev. B 6, (2000) 1278.