- Email: [email protected]
Micromagnetic study of reversible transverse susceptibility
Physica B 306 (2001) 221–227
Micromagnetic study of reversible transverse susceptibility L. Spinua,*, Al. Stancua,b, C.J. O’Connora a
Advanced Materials Research Institute, University of New Orleans, 2000 Lakeshore drive, New Orleans, LA 70148, USA b Faculty of Physics, ‘‘Alexandru Ioan Cuza’’ University, Iasi 6600, Romania
Abstract A generalized theoretical approach of the transverse susceptibility in the case of uniaxial ferromagnets is presented. This advances the classical model of transverse susceptibility where only the first-order anisotropy constant is taken into account, and makes possible the study of the influence of the second-order anisotropy constant on the transverse susceptibility curves. The complicated switching behavior occurring in certain cases was analyzed and discussed from the dynamical point of view, comparing the results with those obtained with a Landau–Lifshitz–Gilbert approach. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Transverse susceptibility; Uniaxial anisotropy; Micromagnetism
1. Introduction Mentioned for the first time by Gans in 1909 [1], the method of transverse susceptibility (wT ) was studied by many scientists due to its evident advantage over other experimental methods yielding valuable information about magnetic anisotropy in particulate media. In previous theoretical approaches of wT [2] it was considered a system of uniaxial particles with the anisotropy energy described only by the first term of the anisotropy energy development WK ¼ K1 sin2 y þ K2 sin4 y þ ?: This is a main source of systematic errors in determining the value of the anisotropy field HK : The peaks occurring at H ¼ 7HK ; in the field variation of wT ; are due to the assembly’s particles that have their easy axis (EA) perpendi*Corresponding author. Tel.: +1-504-280-1122; fax: +1504-280-3185. E-mail address: [email protected] (L. Spinu).
cular to the applied field direction (c ¼ 901); in this case, y; the orientation of the equilibrium position of the magnetic moment with respect of the EA direction, is close to 901, and the expression WK1 ¼ K1 sin2 y is not anymore a good approximation for WK : In this paper we present a systematic study of the influence of the higher-order terms of the anisotropy energy development WK on wT : A generalized expression for wT is proposed and the field variation of wT ðhÞ is calculated over a wide range of situations encountered experimentally (positive and negative values for k2 ¼ K2 =K1 ). All the calculations are based on a generalized Stoner–Wohlfarth (SW) model using the generalized critical curves method [3]. The complicated switching behavior occurring in certain cases was analyzed and discussed from the dynamical point of view, comparing the results with those obtained with a Landau– Lifshitz–Gilbert (LLG) approach.
0921-4526/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 0 0 8 - 0
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2. Model The transverse susceptibility wT is measured by applying a small perturbing alternating field, HAC ; perpendicular to the main bias magnetic field, HDC ; and detecting the component of magnetization variation along HAC : The first coherent theoretical approach for wT is due to Aharoni [2], who calculated the reversible susceptibility tensor of an SW particle [4], wij : The susceptibility tensor is defined as wij ¼ dMi =dHj and its diagonal components comprise the longitudinal susceptibility component, wL ; measured in the main field bias direction, and the two transverse susceptibility components, wT1 and wT2 ; measured in directions perpendicular to the bias field. We will briefly outline the derivation of the wT expression in the case of a uniaxial single-domain SW particle with volume V; anisotropy constants K1 and K2 ; and saturation magnetization MS : HDC is applied along the z-axis and the HAC along the x-axis (Fig. 1). Based on the coordinate system given in Fig. 1 the energy density of the magnetic system considered is ~S ~ ~S ~ W ¼ K 1 ðM uK Þ 2 K 2 ð M uK Þ 4 ~ S ðH ~DC þ H ~AC Þ; M
ð1Þ
where the first two terms correspond to the anisotropy energy contribution and the last one represents the energy density of both the DC and AC fields. The vector ! u K is the unit vector in the EA direction. In this case, the longitudinal susceptibility is wL ¼ wzz ¼ dMz =dHz with Hx ¼ Hy ¼ 0; and the transverse susceptibility components are wT1 ¼ wxx ¼ ðdMx =dHx ÞHx ¼0 with Hy ¼ 0; and wT2 ¼ wyy ¼ ðdMy =dHy ÞHy ¼0 with Hx ¼ 0: Mi and Hi (i ¼ x; y; z) are the projections of the ~ S ¼ Ms ~ vectors saturation magnetization, M uM ¼ Ms ðsin yM cos jM ; sin yM sin jM ; cos yM Þ; and to~¼H ~DC þ H ~AC ¼ H~ tal magnetic field, H uH ¼ Hðsin yH ; 0; cos yH Þ; respectively (Fig. 1). It follows that 3 dðsin yM cos jM Þ ; wT1 ¼ w0 lim 2 yH -0 dðH sin yH Þ
ð2Þ
where w0 ¼ MS2 =3K1 : The torques caused by the anisotropy and bias magnetic field, HDC ; will determine the equilibrium position of the particle magnetic moment ðyM ; jM Þ; while the small alternating field, HAC ; will produce only small perturbations around this equilibrium position. From the point of view of the energy of the magnetic system, the stable equilibrium positions of magnetization vector are given by the conditions (qW=qyM ¼ 0; q2 W=qy2M X0) and (qW=qjM ¼ 0; q2 W=qj2M X0) which are the conditions for minimum of the total energy. Using a similar perturbation approach as in Ref. [2], in the case of a system with the total energy density described by Eq. (1), for the reversible transverse susceptibility wT1 wT one obtains " 3 wT ¼ w0 cos2 jK 2 cos2 yM h cos yM þ cos 2y þ k2 ðcos 2y cos 4yÞ # sin y 2 sin jK ; ð3Þ h sin yK
Fig. 1. The vector diagram corresponding to a uniaxial magnetic particle subjected to a magnetization process in perpendicular fields.
where h ¼ HDC =HK1 and y ¼ yM yK ; with HK1 ¼ 2K1 =MS : It can be easily seen that for k2 ¼ 0 one obtains the wT as presented in Ref. [2]. In this case, the wT
L. Spinu et al. / Physica B 306 (2001) 221–227
evaluation is simpler due to the fact that the free energy could only have two minima, yM : When higher-order term K2 is present in the uniaxial anisotropy energy, WK ; the nature of the anisotropy energy ground state can change from easyaxis (K1 > 0) or easy-plane (K1 o0) to easy-cone type, depending on the values of the ratio k2 ¼ K2 =K1 : These additional equilibrium positions for the magnetization vector can generate a very complicated energy profile, where metastable states can coexist with the global minimum. The external field may induce transitions between the additional minima that will be reflected in the major hysteresis loop by apparition of small Barkhausen jumps [5]. The proper detection and interpretation of the Barkhausen jumps is very important because it makes possible to determine the anisotropy constants of different orders. As will be shown later, these additional Barkhausen jumps driven by higher-order anisotropy constants emerge more evidently in the field dependence of transverse susceptibility, making their detection much easier than from the major hysteresis loop. Under the conditions of a complex energy profile generated by the presence of higher-order anisotropy constants, finding the field variation of the transverse susceptibility using Eq. (3), which depends on the equilibrium position yM ; can be a difficult task. In order to select the appropriate stable equilibrium orientation yM ; a critical curve approach [3] was used. The critical curves are determined by the equation system derived from the stable equilibrium condition, qW=qyM ¼ 0; q2 W=qy2M X0: At the critical point q2 W=qy2M ¼ 0; and the locus of critical points in the ðH> ; H8 Þ ! plane (H> is the H component perpendicular to the EA direction and H8 is the component along EA) is given by a set of parametric equations H> ¼ H> ðyM ; K1 ; K2 Þ and H8 ¼ H8 ðyM ; K1 ; K2 Þ: Using the energy density expression (1) for the parametric equations one obtains H> h> ¼ 1 ¼ sin3 y½1 þ k2 ð6sin2 y 4Þ ; HK H8 h8 ¼ 1 ¼ cos3 y½1 þ k2 ð6cos2 y þ 6Þ : ð4Þ HK The critical curves for K1 > 0 and different values of k2 are shown in Fig. 2. As in the case
223
of well-known astroid (for k2 ¼ 0 in Eq. (4)) the magnetization vector stable orientations can be found by construction of proper tangents to the critical curve. The critical curve orientation given by the arrows in Fig. 2 indicates the stable half tangents that correspond to the stable equilibrium orientations yM [3]. One observes that as positive k2 increases, the critical curve extends along the h> direction and for high enough k2 values an extra cusp appears along the h8 direction. A similar behavior, but in an opposite direction, is observed for negative k2 values: a contraction of the critical curve along h> and extra-cusp along the same direction. The change in the shape of the critical curve and especially the occurrence of the extracusps is very important because they determine a qualitative change in the switching behavior of the system. The direction of the applied field will cross the critical curve in points that correspond to field values where Barkhausen jumps occur. For orientations of the applied field that cross the critical curve extra-cusps, additional Barkhausen jumps will occur (see Fig. 5). In order to find the new stable orientation of magnetization after jump, one has to travel along the critical curve in the sense given by its orientation up to a point where the stable half tangent passes through the jump point. From the parametric equations of the critical curves (4) one can easily determine the conditions of the appearance of the extra-cusps along h> and h8 : So, from the condition h8 ¼ 0 in the interval ð0; p=2Þ we have two solutions, the expected y ¼ 0 and the additional one sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4k2 1 y ¼ arcsin 6k2 that exists only for 4k2 1 > 0: Therefore, a cusp appears along the h> -axis only for anisotropy constants that satisfy the conditions 4K2 > K1 and K1 > 0: Analogously, from h> ¼ 0 the conditions K1 > 6K2 and K1 > 0 of emergence of the extracusp along the h> -axis can be derived. A negative value for K1 corresponds to an inversion of both h> - and h8 -axis (negative sign in Eq. (4)) and it generates a reversing of critical curve orientation. It can be shown that for positive and negative
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Fig. 2. The critical curves for K1 > 0 and different values of k2 :
values of K1 and K2 there are eight different configurations for the critical curves [6].
3. Results First we consider the case of a single domain particle with a uniaxial magneto-crystalline anisotropy free energy with K1 > 0; and different positive values for k2 : In this case the anisotropy axis is an easy magnetization direction, and different values of k2 will affect only the shape of the critical curve. The calculated wT curves for different orientations yK of the applied field with respect to the EA, as the field is swept from positive saturation to negative one, are presented in the panel of Fig. 3. For yK ¼ 01 we notice that wT is the same for all values of k2 Að0; 14Þ; giving the expected Barkhausen jump at h ¼ H=HK1 ¼ 1:
Only for k2 > 14; wT presents an additional singularity located at a reduced field value depending on k2 ; hðk2 Þ ¼ 4k2 ðð1 þ 2k2 Þ=6k2 Þ3=2 ; due to the appearance of the cusps in the EA direction. For yK ¼ 901; wT is symmetric with two singularities located at hðk2 Þ ¼ 7ð1 þ 2k2 Þ: For all the other orientations of the applied field, the effect of a positive k2 is a displacement of the anisotropy and switching peaks, according to the shape change of the modified astroid. Fig. 4 summarizes the results obtained for K1 > 0 and different negative values for k2 : For yK ¼ 01; wT curve does not depend on k2 ; and is identical to that for k2 ¼ 0: At yK ¼ 901 and k2 Að16; 0Þ; symmetric peaks located at hðk2 Þ ¼ 7ð1 þ 2k2 Þ are observed. For k2 o 16 we have two different types of behavior: for k2 Að23; 16Þ one obtains two asymmetric peaks located at hðk2 Þ ¼ þð1 þ 2k2 Þ and hðk2 Þ ¼ 4k2 ð1=6k2 Þ3=2 ; and for k2 o 23; wT
L. Spinu et al. / Physica B 306 (2001) 221–227
225
Fig. 3. The theoretical curves of wT for a mono-domain particle having a uniaxial anisotropy energy with K1 > 0; and different positive values for k2 : The different figures in the panel corresponds to different values for yK :
Fig. 4. The theoretical curves of wT for a mono-domain particle having an uniaxial anisotropy energy with K1 > 0 and K2 o0: The different figures in the panel correspond to different values for yK :
presents a single central peak. For yK ¼ 301 and k2 sufficiently negative, instead of one switching peak per sweep one obtains two switching peaks. The
additional Barkhausen jump in this case is the direct consequence of the second anisotropy constant. The panel of Fig. 5 presents in detail
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Fig. 5. The calculated hysteresis and wT curves for k2 ¼ 1; K1 > 0 and yK ¼ 301: The right panel shows the critical curve and the direction of applied field in the same conditions.
this case, for k2 ¼ 1 showing in addition to the wT curve, the hysteresis loop and the critical curve. Note the points where the Barkhausen jumps occur in all the curves. We mention that in Ref. [6], the occurrence of the two switching peaks was not remarked in this particular case. This complex switching behavior was confirmed by micromagnetic calculations of both hysteresis loop and wT ; based on an LLG approach [7]. The LLG results are displayed in Fig. 5 by open symbols. At yK ¼ 601 we notice only the reversible behavior on the wT curve for k2 o 23: This reversible behavior occurs in a wider angular domain for yK than presented by Thiaville in Ref. [3], as confirmed by micromagnetic LLG calculations. The easy-plane and easy-cone symmetries for magneto-crystalline free energy can be treated with the same uniaxial anisotropy approach using a negative value for K1 : Hence, for K1 o0 and k2 o12 the system has an easy plane and a reversible and symmetric behavior is obtained on wT curves. For
k2 > 12; an easy cone replaces the easy plane and a more complex peak structure for wT is observed.
4. Conclusions A comprehensive theoretical approach of the transverse susceptibility, which accounts for the higher-order anisotropy constants was presented. It is shown that using the transverse susceptibility method it is possible to detect the Barkhausen jumps driven by higher-order anisotropy constants, which allows their determination directly.
Acknowledgements Work at AMRI was supported through DARPA grant No. MDA 972-97-1-0003. A.S. acknowledges the support from CNCSIS grants A and D. L.S. acknowledges useful discussions with H. Srikanth.
L. Spinu et al. / Physica B 306 (2001) 221–227
References [1] G. Gans, Ann. Phys. 29 (1909) 301. [2] A. Aharoni, et al., Bull. Res. Counc. Israel 6A (1957) 215. [3] A. Thiaville, J. Magn. Magn. Mater. 182 (1998) 5.
227
[4] E.C. Stoner, E.P. Wolfarth, Philos. Trans. Roy. Soc. London Ser. A 240 (1948) 599. [5] G. Bertotti, Hysteresis and Magnetism, Academic Press, San Diego, 1998, p. 245. [6] C.-R. Chang, J. Appl. Phys. 69 (1991) 2431. [7] P.R. Gillete, K. Oshima, J. Appl. Phys. 29 (1958) 529 .
Micromagnetic study of reversible transverse susceptibility L. Spinua,*, Al. Stancua,b, C.J. O’Connora a
Advanced Materials Research Institute, University of New Orleans, 2000 Lakeshore drive, New Orleans, LA 70148, USA b Faculty of Physics, ‘‘Alexandru Ioan Cuza’’ University, Iasi 6600, Romania
Abstract A generalized theoretical approach of the transverse susceptibility in the case of uniaxial ferromagnets is presented. This advances the classical model of transverse susceptibility where only the first-order anisotropy constant is taken into account, and makes possible the study of the influence of the second-order anisotropy constant on the transverse susceptibility curves. The complicated switching behavior occurring in certain cases was analyzed and discussed from the dynamical point of view, comparing the results with those obtained with a Landau–Lifshitz–Gilbert approach. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Transverse susceptibility; Uniaxial anisotropy; Micromagnetism
1. Introduction Mentioned for the first time by Gans in 1909 [1], the method of transverse susceptibility (wT ) was studied by many scientists due to its evident advantage over other experimental methods yielding valuable information about magnetic anisotropy in particulate media. In previous theoretical approaches of wT [2] it was considered a system of uniaxial particles with the anisotropy energy described only by the first term of the anisotropy energy development WK ¼ K1 sin2 y þ K2 sin4 y þ ?: This is a main source of systematic errors in determining the value of the anisotropy field HK : The peaks occurring at H ¼ 7HK ; in the field variation of wT ; are due to the assembly’s particles that have their easy axis (EA) perpendi*Corresponding author. Tel.: +1-504-280-1122; fax: +1504-280-3185. E-mail address: [email protected] (L. Spinu).
cular to the applied field direction (c ¼ 901); in this case, y; the orientation of the equilibrium position of the magnetic moment with respect of the EA direction, is close to 901, and the expression WK1 ¼ K1 sin2 y is not anymore a good approximation for WK : In this paper we present a systematic study of the influence of the higher-order terms of the anisotropy energy development WK on wT : A generalized expression for wT is proposed and the field variation of wT ðhÞ is calculated over a wide range of situations encountered experimentally (positive and negative values for k2 ¼ K2 =K1 ). All the calculations are based on a generalized Stoner–Wohlfarth (SW) model using the generalized critical curves method [3]. The complicated switching behavior occurring in certain cases was analyzed and discussed from the dynamical point of view, comparing the results with those obtained with a Landau– Lifshitz–Gilbert (LLG) approach.
0921-4526/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 0 0 8 - 0
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L. Spinu et al. / Physica B 306 (2001) 221–227
2. Model The transverse susceptibility wT is measured by applying a small perturbing alternating field, HAC ; perpendicular to the main bias magnetic field, HDC ; and detecting the component of magnetization variation along HAC : The first coherent theoretical approach for wT is due to Aharoni [2], who calculated the reversible susceptibility tensor of an SW particle [4], wij : The susceptibility tensor is defined as wij ¼ dMi =dHj and its diagonal components comprise the longitudinal susceptibility component, wL ; measured in the main field bias direction, and the two transverse susceptibility components, wT1 and wT2 ; measured in directions perpendicular to the bias field. We will briefly outline the derivation of the wT expression in the case of a uniaxial single-domain SW particle with volume V; anisotropy constants K1 and K2 ; and saturation magnetization MS : HDC is applied along the z-axis and the HAC along the x-axis (Fig. 1). Based on the coordinate system given in Fig. 1 the energy density of the magnetic system considered is ~S ~ ~S ~ W ¼ K 1 ðM uK Þ 2 K 2 ð M uK Þ 4 ~ S ðH ~DC þ H ~AC Þ; M
ð1Þ
where the first two terms correspond to the anisotropy energy contribution and the last one represents the energy density of both the DC and AC fields. The vector ! u K is the unit vector in the EA direction. In this case, the longitudinal susceptibility is wL ¼ wzz ¼ dMz =dHz with Hx ¼ Hy ¼ 0; and the transverse susceptibility components are wT1 ¼ wxx ¼ ðdMx =dHx ÞHx ¼0 with Hy ¼ 0; and wT2 ¼ wyy ¼ ðdMy =dHy ÞHy ¼0 with Hx ¼ 0: Mi and Hi (i ¼ x; y; z) are the projections of the ~ S ¼ Ms ~ vectors saturation magnetization, M uM ¼ Ms ðsin yM cos jM ; sin yM sin jM ; cos yM Þ; and to~¼H ~DC þ H ~AC ¼ H~ tal magnetic field, H uH ¼ Hðsin yH ; 0; cos yH Þ; respectively (Fig. 1). It follows that 3 dðsin yM cos jM Þ ; wT1 ¼ w0 lim 2 yH -0 dðH sin yH Þ
ð2Þ
where w0 ¼ MS2 =3K1 : The torques caused by the anisotropy and bias magnetic field, HDC ; will determine the equilibrium position of the particle magnetic moment ðyM ; jM Þ; while the small alternating field, HAC ; will produce only small perturbations around this equilibrium position. From the point of view of the energy of the magnetic system, the stable equilibrium positions of magnetization vector are given by the conditions (qW=qyM ¼ 0; q2 W=qy2M X0) and (qW=qjM ¼ 0; q2 W=qj2M X0) which are the conditions for minimum of the total energy. Using a similar perturbation approach as in Ref. [2], in the case of a system with the total energy density described by Eq. (1), for the reversible transverse susceptibility wT1 wT one obtains " 3 wT ¼ w0 cos2 jK 2 cos2 yM h cos yM þ cos 2y þ k2 ðcos 2y cos 4yÞ # sin y 2 sin jK ; ð3Þ h sin yK
Fig. 1. The vector diagram corresponding to a uniaxial magnetic particle subjected to a magnetization process in perpendicular fields.
where h ¼ HDC =HK1 and y ¼ yM yK ; with HK1 ¼ 2K1 =MS : It can be easily seen that for k2 ¼ 0 one obtains the wT as presented in Ref. [2]. In this case, the wT
L. Spinu et al. / Physica B 306 (2001) 221–227
evaluation is simpler due to the fact that the free energy could only have two minima, yM : When higher-order term K2 is present in the uniaxial anisotropy energy, WK ; the nature of the anisotropy energy ground state can change from easyaxis (K1 > 0) or easy-plane (K1 o0) to easy-cone type, depending on the values of the ratio k2 ¼ K2 =K1 : These additional equilibrium positions for the magnetization vector can generate a very complicated energy profile, where metastable states can coexist with the global minimum. The external field may induce transitions between the additional minima that will be reflected in the major hysteresis loop by apparition of small Barkhausen jumps [5]. The proper detection and interpretation of the Barkhausen jumps is very important because it makes possible to determine the anisotropy constants of different orders. As will be shown later, these additional Barkhausen jumps driven by higher-order anisotropy constants emerge more evidently in the field dependence of transverse susceptibility, making their detection much easier than from the major hysteresis loop. Under the conditions of a complex energy profile generated by the presence of higher-order anisotropy constants, finding the field variation of the transverse susceptibility using Eq. (3), which depends on the equilibrium position yM ; can be a difficult task. In order to select the appropriate stable equilibrium orientation yM ; a critical curve approach [3] was used. The critical curves are determined by the equation system derived from the stable equilibrium condition, qW=qyM ¼ 0; q2 W=qy2M X0: At the critical point q2 W=qy2M ¼ 0; and the locus of critical points in the ðH> ; H8 Þ ! plane (H> is the H component perpendicular to the EA direction and H8 is the component along EA) is given by a set of parametric equations H> ¼ H> ðyM ; K1 ; K2 Þ and H8 ¼ H8 ðyM ; K1 ; K2 Þ: Using the energy density expression (1) for the parametric equations one obtains H> h> ¼ 1 ¼ sin3 y½1 þ k2 ð6sin2 y 4Þ ; HK H8 h8 ¼ 1 ¼ cos3 y½1 þ k2 ð6cos2 y þ 6Þ : ð4Þ HK The critical curves for K1 > 0 and different values of k2 are shown in Fig. 2. As in the case
223
of well-known astroid (for k2 ¼ 0 in Eq. (4)) the magnetization vector stable orientations can be found by construction of proper tangents to the critical curve. The critical curve orientation given by the arrows in Fig. 2 indicates the stable half tangents that correspond to the stable equilibrium orientations yM [3]. One observes that as positive k2 increases, the critical curve extends along the h> direction and for high enough k2 values an extra cusp appears along the h8 direction. A similar behavior, but in an opposite direction, is observed for negative k2 values: a contraction of the critical curve along h> and extra-cusp along the same direction. The change in the shape of the critical curve and especially the occurrence of the extracusps is very important because they determine a qualitative change in the switching behavior of the system. The direction of the applied field will cross the critical curve in points that correspond to field values where Barkhausen jumps occur. For orientations of the applied field that cross the critical curve extra-cusps, additional Barkhausen jumps will occur (see Fig. 5). In order to find the new stable orientation of magnetization after jump, one has to travel along the critical curve in the sense given by its orientation up to a point where the stable half tangent passes through the jump point. From the parametric equations of the critical curves (4) one can easily determine the conditions of the appearance of the extra-cusps along h> and h8 : So, from the condition h8 ¼ 0 in the interval ð0; p=2Þ we have two solutions, the expected y ¼ 0 and the additional one sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4k2 1 y ¼ arcsin 6k2 that exists only for 4k2 1 > 0: Therefore, a cusp appears along the h> -axis only for anisotropy constants that satisfy the conditions 4K2 > K1 and K1 > 0: Analogously, from h> ¼ 0 the conditions K1 > 6K2 and K1 > 0 of emergence of the extracusp along the h> -axis can be derived. A negative value for K1 corresponds to an inversion of both h> - and h8 -axis (negative sign in Eq. (4)) and it generates a reversing of critical curve orientation. It can be shown that for positive and negative
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Fig. 2. The critical curves for K1 > 0 and different values of k2 :
values of K1 and K2 there are eight different configurations for the critical curves [6].
3. Results First we consider the case of a single domain particle with a uniaxial magneto-crystalline anisotropy free energy with K1 > 0; and different positive values for k2 : In this case the anisotropy axis is an easy magnetization direction, and different values of k2 will affect only the shape of the critical curve. The calculated wT curves for different orientations yK of the applied field with respect to the EA, as the field is swept from positive saturation to negative one, are presented in the panel of Fig. 3. For yK ¼ 01 we notice that wT is the same for all values of k2 Að0; 14Þ; giving the expected Barkhausen jump at h ¼ H=HK1 ¼ 1:
Only for k2 > 14; wT presents an additional singularity located at a reduced field value depending on k2 ; hðk2 Þ ¼ 4k2 ðð1 þ 2k2 Þ=6k2 Þ3=2 ; due to the appearance of the cusps in the EA direction. For yK ¼ 901; wT is symmetric with two singularities located at hðk2 Þ ¼ 7ð1 þ 2k2 Þ: For all the other orientations of the applied field, the effect of a positive k2 is a displacement of the anisotropy and switching peaks, according to the shape change of the modified astroid. Fig. 4 summarizes the results obtained for K1 > 0 and different negative values for k2 : For yK ¼ 01; wT curve does not depend on k2 ; and is identical to that for k2 ¼ 0: At yK ¼ 901 and k2 Að16; 0Þ; symmetric peaks located at hðk2 Þ ¼ 7ð1 þ 2k2 Þ are observed. For k2 o 16 we have two different types of behavior: for k2 Að23; 16Þ one obtains two asymmetric peaks located at hðk2 Þ ¼ þð1 þ 2k2 Þ and hðk2 Þ ¼ 4k2 ð1=6k2 Þ3=2 ; and for k2 o 23; wT
L. Spinu et al. / Physica B 306 (2001) 221–227
225
Fig. 3. The theoretical curves of wT for a mono-domain particle having a uniaxial anisotropy energy with K1 > 0; and different positive values for k2 : The different figures in the panel corresponds to different values for yK :
Fig. 4. The theoretical curves of wT for a mono-domain particle having an uniaxial anisotropy energy with K1 > 0 and K2 o0: The different figures in the panel correspond to different values for yK :
presents a single central peak. For yK ¼ 301 and k2 sufficiently negative, instead of one switching peak per sweep one obtains two switching peaks. The
additional Barkhausen jump in this case is the direct consequence of the second anisotropy constant. The panel of Fig. 5 presents in detail
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L. Spinu et al. / Physica B 306 (2001) 221–227
Fig. 5. The calculated hysteresis and wT curves for k2 ¼ 1; K1 > 0 and yK ¼ 301: The right panel shows the critical curve and the direction of applied field in the same conditions.
this case, for k2 ¼ 1 showing in addition to the wT curve, the hysteresis loop and the critical curve. Note the points where the Barkhausen jumps occur in all the curves. We mention that in Ref. [6], the occurrence of the two switching peaks was not remarked in this particular case. This complex switching behavior was confirmed by micromagnetic calculations of both hysteresis loop and wT ; based on an LLG approach [7]. The LLG results are displayed in Fig. 5 by open symbols. At yK ¼ 601 we notice only the reversible behavior on the wT curve for k2 o 23: This reversible behavior occurs in a wider angular domain for yK than presented by Thiaville in Ref. [3], as confirmed by micromagnetic LLG calculations. The easy-plane and easy-cone symmetries for magneto-crystalline free energy can be treated with the same uniaxial anisotropy approach using a negative value for K1 : Hence, for K1 o0 and k2 o12 the system has an easy plane and a reversible and symmetric behavior is obtained on wT curves. For
k2 > 12; an easy cone replaces the easy plane and a more complex peak structure for wT is observed.
4. Conclusions A comprehensive theoretical approach of the transverse susceptibility, which accounts for the higher-order anisotropy constants was presented. It is shown that using the transverse susceptibility method it is possible to detect the Barkhausen jumps driven by higher-order anisotropy constants, which allows their determination directly.
Acknowledgements Work at AMRI was supported through DARPA grant No. MDA 972-97-1-0003. A.S. acknowledges the support from CNCSIS grants A and D. L.S. acknowledges useful discussions with H. Srikanth.
L. Spinu et al. / Physica B 306 (2001) 221–227
References [1] G. Gans, Ann. Phys. 29 (1909) 301. [2] A. Aharoni, et al., Bull. Res. Counc. Israel 6A (1957) 215. [3] A. Thiaville, J. Magn. Magn. Mater. 182 (1998) 5.
227
[4] E.C. Stoner, E.P. Wolfarth, Philos. Trans. Roy. Soc. London Ser. A 240 (1948) 599. [5] G. Bertotti, Hysteresis and Magnetism, Academic Press, San Diego, 1998, p. 245. [6] C.-R. Chang, J. Appl. Phys. 69 (1991) 2431. [7] P.R. Gillete, K. Oshima, J. Appl. Phys. 29 (1958) 529 .