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Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]]]]
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7 Micromagnetic structures in square magnetic nanodots
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M. Maicasa, M.A. Riveroa, E. L!opeza,*, M.C. S!ancheza, C. Arocab, P. S!anchezb
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a Departmento de F!isica de Materiales, Facultad de CC. F!isicas, Univ. Complutense, 28040 Madrid, Spain
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13 I.S.O.M., E.T.S.I. Telecomunicaci!on, Univ. Polit!ecnica, 28040 Madrid, Spain
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Abstract
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The structure of magnetization in magnetic square dots is studied in the range of tens of nanometer by means of
19 micromagnetic simulations. Two magnetic configurations are found, diagonal and vortex structures for thin films, flower
and vortex for magnetic nanocubes and a hybrid flower in bulk and vortex near the surfaces for dots with thicknesses
21 larger than dot edge length. r 2002 Published by Elsevier Science B.V.
23 Keywords: Micromagnetism; Nanodot; Magnetostatic; Simulation
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The rapidly decreasing size of magnetic devices has 100 erg=cm3 parallel to the x-axis has also been
27 increased the interest in magnetic structures for particles considered. The criterion for convergence used was a 57
at nanoscale dimensions. Nanoimprint lithography can maximum variation in magnetization director cosines
29 result in magnetic nanostructures which seem applicable smaller than 2 10 4 and the grid cells used were cubes 59
for ultrahigh recording media [1]. For such high of 3:125 nm edge length. These latter parameters were
31 densities perpendicular recording seems to be appro- used in all calculations in order to keep the same error 61
priate [2]. Magnetic dots with a thickness higher than when making comparisons.
33 dot base dimensions exhibit perpendicular magnetic While anisotropy can play an important role in 63
orientation. This makes an interesting review of the magnetic distribution in dots with dimensions in the
35 magnetic layer thickness influence on magnetic struc- range of hundreds of nanometers [5] at the scale of tens 65
tures for magnetic dots. of nanometers, magnetization distribution appears
37 Micromagnetic simulations prove to be specially mainly dependent on exchange and magnetostatic 67
appropriate for nanoparticles. Calculations with cell interactions. From the exchange point of view, magne-
39 dimensions over the exchange length may result in tization tries to get nearly parallel in order to reduce the 69
computational errors due to an undervaluated exchange exchange contribution. From the magnetostatic point of
41 energy. On the other hand, magnetostatic energy view, magnetization tries to avoid surface poles. The 71
evaluation becomes very time consuming for grids with smaller the sample is, the more important the exchange
43 a large number of cells. At nanoscale dimensions we can becomes and so dots of few tens of nanometers exhibit 73
work with grids with not a large number of cells, keeping nearly parallel magnetic configurations. For bigger dots
45 cell dimensions below the exchange length. Calculations magnetization tries to avoid surface poles and dots 75
have been performed following a Labonte scheme [3] exhibit circular rotating magnetization configurations.
47 minimizing exchange, anisotropy and magnetostatic Certain transition dimensions appear for which these 77
energies. The evaluation of magnetostatic energy is two different structures have nearly the same energy.
49 accelerated 79
51 81
53
*Correspo UNCORRECTED PROOF
by means of the fast Fourier transform For thin film magnetic dots calculations lead to two
technique [4]. Parameters used in the calculations are different configurations as shown in Fig. 1 [6,7], one
typical for permalloy Ms ź 800 emu=cm3; Aex ź 1:3 with magnetization parallel to the square diagonal,
10 6 erg=cm and an uniaxial anisotropy of K ź diagonal, and another with rotating magnetization,
vortex. A third structure can be reached with magnetiza- 83
nding author. Tel.: +34-91-3944549; fax: +34-91- tion parallel to dot edge although with higher energy
55 3944547. than that with magnetization parallel to the dot 85
E-mail address: elolopez@eucmax.sim.ucm.es (E. L!opez).
0304-8853/01/$ - see front matter r 2002 Published by Elsevier Science B.V.
PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 1 3 5 7 - 9
MAGMA : 8775
ARTICLE IN PRESS
2 M. Maicas et al. / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]]]]
1 Table 2 49
Magnetic energy in 50 50 nm dots for different thicknesses
3 51
Thickness (nm) Diagonal (erg) Vortex (erg)
5 6 6:6 10 12 No convergence 53
12 2:0 10 11 2:7 10 11
7 25 5:8 10 11 5:2 10 11 55
50 9:9 10 11 9:9 10 11
9 100 No convergence 1:7 10 10 57
11 59
Fig. 1. (a) diagonal and (b) vortex magnetization configura-
13 tions in a 50 50 nm base, 12:5 nm height magnetic dot. 61
15 63
Table 1
17 Magnetic energy in a 10 nm thick dot for different dot 65
dimensions
19 67
Dot edge (nm) Diagonal (erg) Vortex (erg)
21 25 5:6 10 12 No convergence (a) (b) 69
50 1:4 10 11 2:1 10 11
23 100 3:3 10 11 3:0 10 11 71
200 6:9 10 11 4:3 10 11
25 73
27 diagonal. Table 1 shows the magnetic energy for 75
different dot dimensions. The smaller the dot, the more
29 stable is the diagonal state, while the bigger the sample 77
the more stable is the vortex state. (c)
31 The effect of the thickness on the magnetic energy 79
Fig. 2. Magnetization in (a) bottom, (b) middle and (c) top
figures can be seen in Table 2. The diagonal state layers in a 50 50 nm base, 100 nm height magnetic dot.
33 becomes more stable for thinner sample. No conver- 81
gence is even reached for the vortex state for very small
35 thicknesses. For the cube, the diagonal state dissapears 83
in favor of the flower state [8]. It can be seen that the References
37 energy of the flower and vortex states are very similar for 85
a nanocube of 50 nm edge length. For a thicker sample a [1] L. Torres, et al., J. Appl. Phys. 85 (8) (1999) 6208.
39 new magnetic distribution appears as a mix of the flower [2] L. Abelmann, et al., J. Appl. Phys. 87 (9) (2000) 5538. 87
and vortex structures. Fig. 2 shows the magnetic layout [3] A.E. LaBonte, J. Appl. Phys. 40 (1969) 2450.
41 in the lower, middle and upper layers in a 50 50 nm2 [4] M. Mansuripur, R. Giles, IEEE Trans. Magn. 24 (1988) 89
base, 100 nm height magnetic dot. In this case the 2326.
43 magnetization in the center is parallel to the large axis [5] R.P. Cowburn, J. Phys. D: Appl. Phys. 33 (2000) R1R16. 91
while upper and lower layers exhibit a vortex-like [6] J. Miltat, et al., Proceedings of the Congress on Trends in
45 structure with the same distribution but opposite nanotechnology, Toledo, Spain, 2000. 93
directions. [7] M.A. Rivero, et al., Proceedings of the Congress on Trends
47 This work was partially supported by CICYT projects in nanotechnology, Toledo, Spain, 2000.
[8] M.E. Schabes, H.N. Bertram, J. Appl. Phys. 64 (3) (1988) 95
MAT98-0824-C02 and MAT2000-0330-P4-03. 1347.
UNCORRECTED PROOF