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JOURNAL OF APPLIED PHYSICS

VOLUME 89, NUMBER 1

1 JANUARY 2001

Transition from a single-domain to a multidomain state in mesoscopic ferromagnetic Co structures E. Seynaeve,a) G. Rens, A. V. Volodin, K. Temst, C. Van Haesendonck, and Y. Bruynseraede Laboratorium voor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium

共Received 10 April 2000; accepted for publication 19 September 2000兲 We have performed magnetic force microscopy measurements on isolated 35 nm thick rectangular Co structures. The structures have a length L ranging between 0.25 and 10 ␮m and a width W ranging between 0.25 and 5.5 ␮m, covering aspect ratios m⫽L/W between 1 and 40. This enables us to map the transition from a magnetic single-domain state towards a magnetic multidomain state when increasing the size of the structures. This transition depends on the size as well as the aspect ratio of the structures. Our results can be interpreted in terms of the theoretical model developed by A. Aharoni 关J. Appl. Phys. 63, 5879 共1988兲兴. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1324687兴

I. INTRODUCTION

done in 1:1 MIKB:1-propanol for 105 s, followed by submersion in 1-propanol for 30 s. A postdevelopment bake was performed for 90 s at 95 °C. In the next step, 35 nm of Co and a 4 nm top layer of a nonmagnetic material 共Au or Ge兲 were deposited in a molecular beam epitaxy system. The top layer prevents oxidation of the Co layer and is assumed not to influence the magnetic state of the structures. The magnetic images were obtained with a commercial atomic force microscope 共AFM兲 system 共Park Scientific Instruments, M5 Autoprobe兲 operating in the noncontact mode using commercially available Si cantilevers. For the MFM measurements the Si tips of the cantilevers are coated with a Co/Au multilayer.16 A Co reference film, evaporated simultaneously with the submicrometer structures, was determined by x-ray diffraction to be polycrystalline. After liftoff, there remained a topographical feature at the border of some structures. We believe that this feature consists of resist residue, and it will therefore not influence the magnetic measurements.2 It does force us, however, to take MFM images at somewhat larger distances from the sample surface, reducing the magnetic resolution. The separation between sample surface and tip varied between 40 and 60 nm. The rectangular structures were designed such that the transition from the magnetic multidomain to single-domain state could be observed in detail. We kept the Co thickness constant and varied the length L from 0.25 to 10 ␮m and the width W from 0.25 to 5.5 ␮m, covering aspect ratios m ⫽L/W from 1 up to 40.

Recently, there has been considerable interest in ferromagnetic structures with submicrometer lateral dimensions inducing a single-domain configuration.1–12 These structures hold interesting promise for high-density magnetic recording media. Already in 1930, Frenkel and Dorfman13 predicted that, below a critical dimension, ferromagnetic particles would reveal a stable single-domain configuration in zero field. Later, Brown14 calculated an upper and lower bound for the critical dimensions of spherical particles. In 1988, Aharoni15 obtained rigorous lower bounds for ellipsoidal particles. For an ellipsoidal particle with minor semi axis a, a uniform magnetization results in the lowest free energy when a⬍a c0 . The critical dimension a c0 is dependent on the strength of the exchange interaction and the saturation magnetization as well as on the aspect ratio m, the ratio between the major and minor semi axes. This provides the possibility to tune small magnetic structures into a single-domain state by changing their aspect ratio m, i.e., by changing their shape anisotropy. Here, we will present a systematic study of the single-domain to multidomain transition for small rectangular Co structures using magnetic force microscopy 共MFM兲. II. EXPERIMENTAL SETUP

The rectangular magnetic structures were defined in thin Co films using standard electron beam 共e-beam兲 lithography. We used a bilayer resist consisting of a 150 nm bottom layer of poly共methyl methacrylate-methacrylic acid兲 共PMMA/ MAA兲 and a 170 nm top layer of 950 kMol PMMA on a Si/SiO2 共400 nm兲 substrate. The e-beam lithography was performed with a modified JEOL JSM-6400 scanning electron microscope using Proxy–Writer from Raith GmbH. The structures were exposed at a voltage of 40 kV with currents between 20 and 300 pA. After exposure, development was

III. MFM RESULTS

Figure 1 shows a typical MFM picture of our rectangular Co structures. The influence of the topography is negligible due to the large distance between the sample and the cantilever. Two different magnetic configurations can be distinguished. The first one consists of a white and a black spot, while the second configuration exhibits a pattern of lines. These lines reflect the presence of domain wall boundaries

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© 2001 American Institute of Physics

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J. Appl. Phys., Vol. 89, No. 1, 1 January 2001

FIG. 1. Typical MFM picture (50⫻50 ␮m2) of rectangular Co structures with different aspect ratios. The transition from multidomain to single domain at larger aspect ratios is clearly visible. The structure indicated by a white dashed box is a multidomain structure surrounded by single-domain structures.

between differently oriented magnetic domains, indicating that the second configuration is magnetically multidomain.10 The first configuration corresponds to a single-domain structure with the direction of the magnetization oriented parallel to the long axis. The clearly distinguishable black and white spots are caused by the interaction between the MFM tip and the magnetic stray field leaving the sample. When the magnetic stray field leaving the structure is parallel to the magnetization vector in the MFM tip, the tip is attracted towards the sample, resulting in a black spot. When the magnetic stray field is antiparallel, the MFM tip will be repelled and a white spot appears. While the transition from the multidomain to the single-domain state is clearly governed by the aspect ratio and size, the actual magnetic state of the structures close to the transition may be influenced by other parameters. This can be seen in the upper left part of Fig. 1, where a multidomain structure, enclosed by a white dashed line, is surrounded by single-domain structures. We believe that this may be due to irregularities in the shape of the structure, leading to variations in the magnetic microstructure.12 Figure 2共a兲 shows a more detailed view of the transition from multidomain to single domain. The structure on the left 共with lateral dimensions 2⫻10 ␮ m2兲 is clearly in a multidomain state. The structure on the right (1.5⫻10 ␮ m2) is in the single-domain state. One can see from the corresponding AFM topography in Fig. 2共b兲 that the geometry of the structure is well defined. In order to minimize the influence of other parameters, e.g., changes in the polycrystalline microstructure, a number of pictures similar to Figs. 1 and 2 have been taken on different samples. This allows us to perform a statistical analysis of the transition. Our results confirm that the transition

FIG. 2. 共a兲 15⫻15 ␮ m2 MFM picture illustrating the transition from a single-domain to a multidomain state by changing the aspect ratio. The structure on the left hand side measures 2⫻10 ␮ m2 while the structure on the right hand side measures 1.5⫻10 ␮ m2. 共b兲 Topographical AFM image of the structures shown in 共a兲. The height of the structure is 35 nm.

from a single-domain to a multidomain state depends both on the size and the aspect ratio of the rectangular structures. Figure 3 shows the results of our statistical analysis. We have plotted the aspect ratio m versus the length L of the structures. The symbols indicate in which magnetic state the particles have been found. Open squares indicate structures in a multidomain state, while closed circles indicate structures that are single domain. Asterisks refer to the mixed occurrence of the two magnetic states in different structures with the same dimensions. The solid line is a fit according to Aharoni15 共see below兲. Measurements made by other groups on Co structures of similar thickness4,9 fit nicely into our diagram.

IV. DISCUSSION

Numerical calculations and Bitter pattern experiments by Wang et al.17 established that the magnetic domain configuration of small multidomain magnetic particles reveals the well known closure domains. As indicated before, the white lines in Figs. 1 and 2共a兲 result from the magnetic stray field

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Seynaeve et al.

J. Appl. Phys., Vol. 89, No. 1, 1 January 2001

FIG. 3. Diagram showing the magnetic state of a structure as function of the aspect ratio m and the length L. Open squares indicate a magnetic multidomain state and closed circles a single-domain state. Asterisks refer to the mixed occurrence of both magnetic states. The full line represents a fit using the theory of Aharoni 共see Ref. 15兲. It is clearly seen that the transition from a single-domain to a multidomain state depends both on the size and the aspect ratio of the rectangular structures.

at the domain wall boundaries. The domain wall configuration for our Co samples is very similar to the results of Wang et al. Our MFM results can be compared to the combined MFM and Lorentz transmission electron microscopy studies by Gomez et al.10 on permalloy structures. Gomez et al. also find patterns of lines in the MFM images which they are able to link to closure domains formed by 90° and 180° domain walls. However, on their Lorentz transmission electron microscopy images, one can see that some structures consist of a mainly single-domain body with ‘‘flux closure’’ ends. Such a configuration would show up on our MFM images as a black and white pair of spots and cannot be distinguished from a pure single-domain configuration due to the increased distance between sample surface and MFM tip, implying a limited lateral resolution. Three main areas can be distinguished in Fig. 3. First, there are the two areas where the rectangular structures are either pure single domain or pure multidomain. In between these two states, there exists a transition regime. The transition from the single-domain to the multidomain state clearly depends on both the size and the aspect ratio of the structures. One can define a critical length L c (m) so that if L ⬍L c , the structure is not in a magnetic multidomain state. Aharoni15 has examined ellipsoidal magnetic particles with a minor semi-axis a. He established that for a⬍a c0 , the critical dimension, these particles will certainly be in a single-domain state. The critical dimension a c0 depends on the aspect ratio m and material constants. Although this result has been derived for ellipsoidal particles, Aharoni suggests that the results should be valid for other structures as well, as long as the shape does not differ too drastically from an ellipsoid. However, his calculations assume a singlecrystalline particle. Although, in principle, additional effects such as crystal anisotropy18 and surface anisotropy19 can be taken into account, quantitative results for arbitrary anisotropy strengths and directions are not available. Our measure-

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ments of the hysteresis loops for large area Co films deposited on oxidized silicon substrates confirm the complete absence of in-plane anisotropy. The polycrystalline film structure apparently results in the elimination of preferential magnetization directions in the Co material. In Fig. 3 the solid line corresponds to the theoretical boundary predicted by Aharoni,15 below which multidomain particles do not occur. The critical dimension L c ⫽2 m a c0 (m) was calculated using Eqs. 共15兲 and 共19兲 in Ref. 15. Taking the saturation magnetization of bulk Co20 1422 emu/cm3, the fit requires an exchange constant of 1.3 ⫻10⫺4 erg/cm, 2 orders of magnitude higher than the reported bulk value for Co.12 This large discrepancy points to the fact that additional mechanisms, including surface anisotropy, surface roughness, and crystalline imperfections18 favor the formation of single-domain structures. We note that recent micromagnetic calculations12 confirm that a polycrystalline microstructure alters the magnetization distribution in the remanent state. One should also consider the possibility that MFM cannot distinguish between a true single-domain state and a state with flux closure ends as reported by Gomez et al.10 The formation of flux closure ends could lead to an overestimation of the size for single-domain structures in our MFM analysis and move the transition between singledomain and multidomain structures towards smaller L. V. CONCLUSIONS

We have been able to map the transition of small rectangular Co structures from a multi-domain to a singledomain magnetic state using MFM. The results agree qualitatively with the predictions made by Aharoni,15 confirming the existence of a critical length below which the structures become single domain. This critical length depends on the aspect ratio as well as on the size of the structures. ACKNOWLEDGMENTS

This work has been supported by the Fund for Scientific Research-Flanders 共FWO兲, as well as by the Flemish Concerted Action 共GOA兲, and the Belgian Inter-University Attraction Poles 共IUAP兲 research programs. One author 共K.T.兲 is a Postdoctoral Research Fellow of the FWO. 1

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