- Email: [email protected]
Approach to saturation in small isotropic spheres
451
Journal of Magnetism and Magnetic Materials 83 (1990) 451-452 North-Holland
APPROACH A. AHARONI
TO SATURATION
IN SMALL ISOTROPIC
a and J.P. JAKUBOVICS
SPHERES
b
* Dept. Electronics, Weizmann Institute, Rehovoth, Israel h Dept. Metallurgy, Parks Road, Oxford OX1 3PH, UK
Cylindrically symmetric magnetization configurations have been computed for small isotropic minimising an infinite
the total energy with no constraint other than imposing initial susceptibility and no hysteresis at any field.
Hysteresis in ferromagnetic materials is mainly caused by anisotropy. Therefore, an isotropic particle should have no hysteresis in small applied fields. Such a single-valued field-dependence has been calculated by Ritz models [1,2], which showed the unusual feature of an infinite initial susceptibility, but it was not clear if that feature was a real effect or just an artifact of those particular models. We have therefore computed the magnetization configurations of small isotropic spheres without assuming any constraints except for a cylindrical symmetry. The method used was to minimise the total energy in various fields applied in the z-direction. Expressions for the energy terms have been given in ref. [3]; in the present case, the anisotropy was zero. The method of minimising the energy was also the same as in previous computations [3,4]. The values for the exchange constant A and saturation magnetization M, were chosen to fit amorphous particles of Fe,,Si,sB,, and are the same as in refs. [1,2]. Fig. 1 shows the minimal energy magnetization configuration in zero applied field, in a sphere with radius R = 20 nm. Along the z-axis, the magnetization is parallel to z, and along the equator, it is nearly parallel to y,
Fig. 1. Magnetization configuration in a quadrant of an isotropic sphere of radius 20 nm in zero applied field. The magnetization in the other three quadrants can be derived in the same way as in ref. (31. 0304-8853,X)/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
that symmetry.
The resulting
ferromagnetic magnetization
spheres by curves show
most of the volume being taken up with the rotation from 2 to y. The average normalised magnetization (mZ) (= (M_.)/M,) is 0.374, which means that a large fraction of IV, is reached in one discontinuous jump, with an infinite initial susceptibility. This jump brings (ml) even closer to 1 in spheres of smaller radii. Thus. in a sphere with R = 15 nm that value is 0.733. These values are close to those found by the Ritz models [1,2]. With increasing applied fields, the central region, in which the magnetization is approximately parallel to the applied field grows, and the magnetization along the equator rotates towards z. Fig. 2 shows (m,) for two different radii as a function of h (= 6H/M,. where H is the applied field, and H and MS are in Sl units note that the definition of h was given incorrectly in ref. [6], where the intended definition was the same as here). It is seen that after *the jump at h = 0, the magnetization varies nearly linearly with h. and seems to approach smoothly to the saturation value at a field that is very close to the theoretical nucleation field h, [5,1] for each particle size. These results are different from those of the Ritz models [1.2], which showed saturation reached well below h,. In order to see the fine details of the approach to saturation, this part of the magnetization curve for R = 20 nm is plotted on an enlarged scale in fig. 3. The
0.01
o’s
h Fig. 2. Average magnetization component parallel to the applied field plotted as a function of h ( = 6H/M,).
452
A. Ahoront, J. P. Jakubouics
/ Saturation
1.00.
0.98.
q E -0.96.
0.94.
x
Fig. 3. The approach to saturation part of fig. 2.
linear approach is very clearly seen in this figure, which means that the results of the Ritz model and of our preliminary computations [6] which have not been completely minimised, do not apply in this region. Similar linearity has also been found for R = 15 nm. The preliminary results [6] seemed to show stable non-uniform magnetization configurations for h > h “. These solutions will presumably be followed by the magnetization when the field is increased, while for a decreasing field the saturated state must still be the only solution. This feature is in accordance with the theoretical (m,) vs. h curve near nucleation [7]. It can readily be shown that this curve has a negative slope at the nucleation field, which implies some hysteresis, of the same general shape as predicted in fig. 1 of ref. [8]. This Potts model prediction for Fe with the field along a [l 1 l] direction is based on an oversimplification which ignores demagnetization. The predicted discontinuity was looked for, but not observed, in Fe single-crystal spheres [8], and in exceptionally good Fe whiskers [9]. A clearer hysteresis of this nature was found in Fe singlecrystal films [lo], but its interpretation is not quite clear. Our figs. 2 and 3 show saturation at h,, without any high-field hysteresis, which does not fit the theoretical negative slope at nucleation. This should mean that either the hysteresis is too small to be detected by this approach, or that there exists another branch of (mL) vs. h, which our method of minimization cannot reach.
in small mtropic
spheres
We have tried to look for this other branch, by starting from different magnetization configurations. At some stages of the minimizations, this sometimes led to what looked like different results for certain values of h, for different starts of the computations. However, they all converged eventually to the same result. A similar high-field hysteresis has also been encountered in three-dimensional, finite-element computations of the magnetization curve of a highly anisotropic sphere [ll]. Some hysteresis was also found [ll] at lower anisotropy in runs with coarse steps, but it disappeared in runs with finer steps. This might indicate that a high field hysteresis exists only for rather high anisotropies, as assumed in [lo], but this is hardly conclusive, because the computations of ref. [ll] are not sufficiently accurate to determine the nucleation field, or the behaviour near zero field. We conclude that this method of minimization over the many parameters of the magnetization in all the quasi-toroids cannot be used for finding different branches, unless one has some indication of what the configuration looks like. Hysteresis near saturation should, thus, be found in some other way.
References [l] A. Aharoni, J. Appl. Phys. 52 (1981) 933. [2] A. Aharoni, J. Appl. Phys. 55 (1984) 1049. [3] A. Aharoni and J.P. Jakubovics, Phil. Mag. B53 (1986) 133. [4] A. Aharoni and J.P. Jakubovics, IEEE Trans. Magn. MAG-24 (1988) 1892. [5] A. Aharoni, J. Appl. Phys. suppl. 30 (1959) 70s. [6] A. Aharoni and J.P. Jakubovics, J. Appl. Phys. 57 (1985) 3526. [7] F. Pu and B. Li. Kexue Tongbao 26 (1981) 207. [8] G.E. Everett, Y.J. Lin and C.J. Tung, J. Appl. Phys. 53 (1982) 1901. [9] S.D. Hanham, 8. Heinrich and A.S. Arrott, J. Appl. Phys. 50 (1979) 2146. [lo] F.J. Rachford, G.A. Prim, J.J. Krebs and K.B. Hathaway. J. Appt. Phys. 53 (1982) 7966. [ll] D.R. Fredkin and T.R. Koehler, IEEE Trans. Magn. MAG-24 (1988) 2362.
Journal of Magnetism and Magnetic Materials 83 (1990) 451-452 North-Holland
APPROACH A. AHARONI
TO SATURATION
IN SMALL ISOTROPIC
a and J.P. JAKUBOVICS
SPHERES
b
* Dept. Electronics, Weizmann Institute, Rehovoth, Israel h Dept. Metallurgy, Parks Road, Oxford OX1 3PH, UK
Cylindrically symmetric magnetization configurations have been computed for small isotropic minimising an infinite
the total energy with no constraint other than imposing initial susceptibility and no hysteresis at any field.
Hysteresis in ferromagnetic materials is mainly caused by anisotropy. Therefore, an isotropic particle should have no hysteresis in small applied fields. Such a single-valued field-dependence has been calculated by Ritz models [1,2], which showed the unusual feature of an infinite initial susceptibility, but it was not clear if that feature was a real effect or just an artifact of those particular models. We have therefore computed the magnetization configurations of small isotropic spheres without assuming any constraints except for a cylindrical symmetry. The method used was to minimise the total energy in various fields applied in the z-direction. Expressions for the energy terms have been given in ref. [3]; in the present case, the anisotropy was zero. The method of minimising the energy was also the same as in previous computations [3,4]. The values for the exchange constant A and saturation magnetization M, were chosen to fit amorphous particles of Fe,,Si,sB,, and are the same as in refs. [1,2]. Fig. 1 shows the minimal energy magnetization configuration in zero applied field, in a sphere with radius R = 20 nm. Along the z-axis, the magnetization is parallel to z, and along the equator, it is nearly parallel to y,
Fig. 1. Magnetization configuration in a quadrant of an isotropic sphere of radius 20 nm in zero applied field. The magnetization in the other three quadrants can be derived in the same way as in ref. (31. 0304-8853,X)/$03.50 (North-Holland)
0 Elsevier Science Publishers
B.V.
that symmetry.
The resulting
ferromagnetic magnetization
spheres by curves show
most of the volume being taken up with the rotation from 2 to y. The average normalised magnetization (mZ) (= (M_.)/M,) is 0.374, which means that a large fraction of IV, is reached in one discontinuous jump, with an infinite initial susceptibility. This jump brings (ml) even closer to 1 in spheres of smaller radii. Thus. in a sphere with R = 15 nm that value is 0.733. These values are close to those found by the Ritz models [1,2]. With increasing applied fields, the central region, in which the magnetization is approximately parallel to the applied field grows, and the magnetization along the equator rotates towards z. Fig. 2 shows (m,) for two different radii as a function of h (= 6H/M,. where H is the applied field, and H and MS are in Sl units note that the definition of h was given incorrectly in ref. [6], where the intended definition was the same as here). It is seen that after *the jump at h = 0, the magnetization varies nearly linearly with h. and seems to approach smoothly to the saturation value at a field that is very close to the theoretical nucleation field h, [5,1] for each particle size. These results are different from those of the Ritz models [1.2], which showed saturation reached well below h,. In order to see the fine details of the approach to saturation, this part of the magnetization curve for R = 20 nm is plotted on an enlarged scale in fig. 3. The
0.01
o’s
h Fig. 2. Average magnetization component parallel to the applied field plotted as a function of h ( = 6H/M,).
452
A. Ahoront, J. P. Jakubouics
/ Saturation
1.00.
0.98.
q E -0.96.
0.94.
x
Fig. 3. The approach to saturation part of fig. 2.
linear approach is very clearly seen in this figure, which means that the results of the Ritz model and of our preliminary computations [6] which have not been completely minimised, do not apply in this region. Similar linearity has also been found for R = 15 nm. The preliminary results [6] seemed to show stable non-uniform magnetization configurations for h > h “. These solutions will presumably be followed by the magnetization when the field is increased, while for a decreasing field the saturated state must still be the only solution. This feature is in accordance with the theoretical (m,) vs. h curve near nucleation [7]. It can readily be shown that this curve has a negative slope at the nucleation field, which implies some hysteresis, of the same general shape as predicted in fig. 1 of ref. [8]. This Potts model prediction for Fe with the field along a [l 1 l] direction is based on an oversimplification which ignores demagnetization. The predicted discontinuity was looked for, but not observed, in Fe single-crystal spheres [8], and in exceptionally good Fe whiskers [9]. A clearer hysteresis of this nature was found in Fe singlecrystal films [lo], but its interpretation is not quite clear. Our figs. 2 and 3 show saturation at h,, without any high-field hysteresis, which does not fit the theoretical negative slope at nucleation. This should mean that either the hysteresis is too small to be detected by this approach, or that there exists another branch of (mL) vs. h, which our method of minimization cannot reach.
in small mtropic
spheres
We have tried to look for this other branch, by starting from different magnetization configurations. At some stages of the minimizations, this sometimes led to what looked like different results for certain values of h, for different starts of the computations. However, they all converged eventually to the same result. A similar high-field hysteresis has also been encountered in three-dimensional, finite-element computations of the magnetization curve of a highly anisotropic sphere [ll]. Some hysteresis was also found [ll] at lower anisotropy in runs with coarse steps, but it disappeared in runs with finer steps. This might indicate that a high field hysteresis exists only for rather high anisotropies, as assumed in [lo], but this is hardly conclusive, because the computations of ref. [ll] are not sufficiently accurate to determine the nucleation field, or the behaviour near zero field. We conclude that this method of minimization over the many parameters of the magnetization in all the quasi-toroids cannot be used for finding different branches, unless one has some indication of what the configuration looks like. Hysteresis near saturation should, thus, be found in some other way.
References [l] A. Aharoni, J. Appl. Phys. 52 (1981) 933. [2] A. Aharoni, J. Appl. Phys. 55 (1984) 1049. [3] A. Aharoni and J.P. Jakubovics, Phil. Mag. B53 (1986) 133. [4] A. Aharoni and J.P. Jakubovics, IEEE Trans. Magn. MAG-24 (1988) 1892. [5] A. Aharoni, J. Appl. Phys. suppl. 30 (1959) 70s. [6] A. Aharoni and J.P. Jakubovics, J. Appl. Phys. 57 (1985) 3526. [7] F. Pu and B. Li. Kexue Tongbao 26 (1981) 207. [8] G.E. Everett, Y.J. Lin and C.J. Tung, J. Appl. Phys. 53 (1982) 1901. [9] S.D. Hanham, 8. Heinrich and A.S. Arrott, J. Appl. Phys. 50 (1979) 2146. [lo] F.J. Rachford, G.A. Prim, J.J. Krebs and K.B. Hathaway. J. Appt. Phys. 53 (1982) 7966. [ll] D.R. Fredkin and T.R. Koehler, IEEE Trans. Magn. MAG-24 (1988) 2362.