Draft Evolution of the Complexity of Magnetic Domain Structure in Nano-Scale Nickel Particl

Evolution of the Complexity of Magnetic Domain Structure in Nano-Scale NickelParticlesGeorge D. Skidmore, Andrew Kunz, E. Dan Dahlberg, and C. E. Campbell Magnetic Microscopy Center, School of Physics and AstronomyUniversity of Minnesota116 Church St. SE, Minneapolis, MN 55455, USAABSTRACTThe micromagnetic structure of nanofabricated nickel particles has beendetermined as a function of particle diameter and thickness using acombination of high-resolution magnetic force microscopy and Landau-Lifshitz-Gilbert micromagnetic simulations. Cylindrical particles werelithographically prepared with diameters in the range 68 - 900 nm, andthickness between 48 and 140 nm. Extensive measurements andsimulations of the 100 nm thick cylinders are reported in this note. At thisthickness, the complexity of observed magnetic states increases as theparticle diameter increases. Single domain structures are found at thesmallest diameters, with two- and three-domain structures appearing atlarger diameters, characterized by in-plane vorticity and in the simplestcase, a vortex ring. At diameters greater than at least 360 nm, themagnetic structure is composed of random stripe domains very similar tothose found in nickel films of the same thickness.INTRODUCTIONNanofabrication methods using electron beam lithography [1] permit the production of small, regular magnetic particles for studies in fundamental magnetism; this may ultimately prove important for developments in magnetic recording and magnetic sensor technology [2]. At the fundamental level these particles are small enough that the number of achievable magnetic states is expected to be small at shorter length scales andincrease as the diameter increases. It should also be expected that the smallest particles should have relatively simple magnetic structure, while the largest particles will be very complex in their structures.The recent development of nanometer scale monopole spike tips for Magnetic Force Microscopy (MFM) [3] permits the mapping of the magnetic field gradient produced by a magnetic specimen at a resolution of approximately 30 nm[4]. This provides information about magnetic structure, which in the present cases is essentially the magnetic structure of the particle surface nearest the MFM probe. At the same time, developments in magnetic simulation and modeling techniques permit the full simulation of the magnetic structure of such particles, including both the surface and the interior[NEED A REFERENCE TO A LLG PAPER(DAN?)]. In this note we report on the use of high resolution MFM measurements to image the magnetic structure of the circular surface of sub-micron nickel cylinders, and the independent simulation of the full magnetic structure of these particles. With no adjustable parameters in the simulations, the simulations confirm the imaged magnetic structure at the surface, and the resultant three-dimensional magnetic structure of the particles is thereby determined.The results reported here are for particles in zero applied magnetic field. Both the MFM and simulation images of the magnetic structure for seven different diameters are shown in Figs 1-7, moving from the large to the small diameters. Simulations of the smallest cylinders, below approximately 80 nm in diameter, show only a quasi-single domain magnetic state, which has a flower pattern for its magnetic structure (Fig. 7); this structure accounts for the MFM images of the surfaces. As the particle diameter is increased new magnetic states with simple vortex patterns are obtained. When the axis of vorticity is perpendicular to the particle axis as in Fig. 6, we term it “in-plane” vorticity.A more complex pattern is the bull’s-eye structure shown in Fig. 2, which is essentially a vortex ring enclosed in a cylinder; the axis of vorticity encircles the particle at the cylinder mid-plane.As is evident, there is good agreement between the simulations and the MFM measurements as to the diameters at which these magnetic structures become possible. It should be noted that these are metastable states, and thus their occurrence depends in partupon the particle’s history. Consequently several different states may be seen in the same particle, as is confirmed in the experiments.We classify the states not only as a function of particle diameter, but also in terms of the stripe period found in unpatterned nickel films[5]. At 100 nm thicknesses the films have a stripe period of approximately 200 nm. A full report on these films will appear elsewhere[6]. At diameters larger than 3/2 the stripe period, the nickel particles begin to take on the stripe domain structure of the films with somewhat random patterns, evidently only constrained by the nearly constant stripe spacing, as shown in Fig. 1. A more detailed description and discussion of the results for the smallest particles comprises the remainder of this note.EXPERIMENTAL SITUATIONThe particles are made by thermally evaporating nickel through a lithographically defined mask; the details of production are outlined elsewhere [7]. The electron-beam lithography used permits several different particle diameters to be made on the same substrate, along with a large area, semi-infinite, “witness” film. The witness films are used to determine the magnetic anisotropy and the stripe domain period as outlined previously [5,6]. As noted above, the particles are then imaged with high resolution MFM. Included in this study are images from several different 100 nm thick samples. These same magnetization patterns were found in samples ranging from 48 nm to 140 nm thick, but a full study for a wide range of diameters was made only for the 100 nm thick particles. The particles probed here are thus 100 nm thick cylinders with diameters ranging from approximately 60 nm to approximately 1000 nm.High resolution MFM qualitatively measures the field gradients produced above and around the particles with approximately 30 nm spatial resolution [4]. Two commercial MFMs available from Digital Instruments were used in the present work[8]. The lift-height for magnetic imaging, using the tapping/lift mode with phase detection, was always 20 nm in the experiments reported here. High-resolution MFM tips [3], each consisting of a ‘monopole spike’ of 300 nm lengths with 7 nm of cobalt as the active layer, were used for all of the images shown here. Images of these tips are shown in Ref.3 and in Ref. 6. These tips provide high resolution imaging with little tip field induced sample perturbation under ambient conditions.To assure that there was no perturbation of the magnetic state, data was always taken twice, with the same tip magnetized in opposite directions so that both north pole and south pole tip magnetizations are used. Some domains, determined to be those with closure (i.e., magnetic moments in the plane of the surface), show strong contrast only when their magnetization lies along the tip field direction; they show weaker contrast when opposing the tip field. Other domains--those determined to be open (i.e., magnetic moments intersecting the plane)--show strong contrast regardless of tip field. Because these distinctions cannot be made with only one image, there must be pairs of images taken with the same tip, on the same particle, with opposite tip magnetization.The particles have been imaged in both the as-deposited state and again after magnetic saturation perpendicular to the substrate plane in order to test the sensitivity of the magnetic structure to the magnetic history of the particle. As will be discussed below, only particles with diameters near the single domain to in-plane vortex transition (i.e., approximately 150 nm diameters) show a difference in frequency of metastable states depending upon this preparation difference.THEORETICAL METHODSExperimentally information about the magnetization of the MFM tip and sample must be assumed to begin to interpret the images taken [9], whereas in simulations, detailed information about the final magnetic state is readily available. In these simulations the cylinders were modeled as an array of identical cubic grains of nickel with diameters near those measured experimentally.The simulations are carried out by numerically solving the Landau-Lifshitz-Gilbert (LLG) equation of motion for the directions of the magnetic moments, m i, of the individual grains comprising the cylinder [10]:d m i/dt = -γ/(1+α2) m i×H i - αγ/(M s)(1+α2)) m i×m i×H iwhere the magnetic field H i at site i is the sum of the dipolar fields due the all other grains, the exchange fields with the neighboring grains, and the uniaxial anisotropy field due to the crystal structure, particle shape, and manufacturing technique. Thus, aside from the grain sizes, there are only four input parameters for the model, three of which are obtained from experiment: the magnetization and exchange coupling of the grains[11], obtained from bulk nickel, and the measured deposition-induced magnetic anisotropy[5,6]. All results presented were simulated with the measured uniaxial anisotropy along the axis of the cylinder, K u = 6 x 105 erg/cc, which is much larger than that of bulk nickel[12]. Both the saturation magnetization moment, M s=484 emu/cc, and the exchange coupling between grains, 10-6 erg/cm, are for bulk Nickel. The fourth parameter is the Gilbert damping parameter α, which is chosen to be unity for maximum damping in order to maximize the rate of relaxation to the metastable states. The LLG simulation program used is commercially available through Michael Scheinfein[13].The combined MFM and simulation data are used to determine which states are most prevalent at different particle diameters. LLG simulations were done at many diameters between 84 and 900 nm. The LLG method is deterministic, since it follows the natural precession of the magnetic moments; consequently, different initial conditions must be used to reach the different metastable states. In our simulations this was accomplished most conveniently by beginning with different random magnetic initial states, similar to the experimental procedure.RESULTSThe various magnetic structures obtained in the experiments and the simulations are shown in the figures. Two MFM images are shown for each of the cylinders, differing only in that they were taken with opposite tip magnetizations for reasons discussed above. The LLG simulation of the same size cylinder is illustrated with a false-color representation of the z-component of the magnetization at the top circular surface of the cylinder; this false color representation is seen to closely resemble the MFM images . Alsoshown are LLG determined magnetization configurations of a vertical slice (i.e., in the plane of the cylinder axis) where magnetic moments are represented by cones. These slices clearly show the magnetization through the thickness of the particle as well as the tendency for closure at the top and bottom surfaces. For the larger cylinders, two slices showing the LLG determined spins at mid-plane and at the top plane are also included. Only every third cone is shown for clarity of presentation.LARGE PARTICLES (Greater than 3/2 Stripe Period)Fig. 1 shows images of micron-sized particles obtained from both MFM and LLG. The large diameter particles allow for many domains, and many different metastable structures, including stripes. No serviceable classification scheme emerges from this analysis in this size range, except that the stripes are of approximately the same width as seen both experimentally and in simulations of 100 nm thick films [5,6].THREE-DOMAIN PARTICLES (3/2 Magnetic Stripe Period)Figs 2-3 show two distinct magnetization patterns in particles of diameter approximately 288 nm. These patterns differ in the direction of the domain walls surrounding a center domain. 288 nm was chosen in order to explore the 3/2 stripe period measured at 100 nm thickness in the thin films[5,6]. If the domain wall encircles the center domain the particle is termed a ‘bull’s-eye’, which is essentially a vortex ring filling the cylinder. If, on the other hand, the domain wall runs in the same direction on either side, the particle is called a ‘three stripe.’In Fig. 2, the two MFM images of the bull’s-eye particle (Figs a and c) are shown alongside the false color LLG image of the magnetization. As can be seen in the mid-plane-slice, there is a domain wall surrounding a centered bubble domain. This permits the domain wall to close on itself, minimizing the magnetic flux and thus the total energy. The main domains are easily identifiable in the vertical slice, as is the main domain wall. Of interest is the attempt at closure that the particle makes radially, resulting in the circularin-plane vortex. A Neel-like domain wall traverses the particle vertically. It is not acomplete Neel wall, since it culminates in an angle 45 degrees from completion, as can be seen in the top slice.The second magnetic structure found in this particle size, shown in Fig. 3, may be termed a three-stripe particle since its appearance resembles the stripe patterns found in the witness films. The discerning feature of this structure, that the domain walls run in the same direction, can be seen in both the vertical-slice and the mid-slice. The magnetization also follows a dual in-plane vortex pattern closing flux along the top and bottom. However, the edge domains in the three stripe structure are larger than in the bull’s-eye structure, and the flux is not as closed as in the latter. The two different closure schemes make for discernible contrast differences in the MFM data. The three-stripe particles show strong signal above every domain, while the bull’s-eye pattern shows a strong signal above the center domain, and a weaker signal on the outside.TWO-DOMAIN PARTICLES (1 Magnetic Stripe Period)Similar work was done for particles of approximately 200 nm diameters. At this size, two main configurations were experimentally identified. They are similar to the above structures and hence we use similar nomenclature. The two-stripe structure, shown in Fig. 4, closes the two main domains across the top in an in-plane vortex fashion. Here, as above, a line followed vertically through the domain wall at the particle center makes a traditional Neel wall turn; unlike that seen above in the bull’s-eye, this Neel wall is much nearer to completion, within 10 degrees instead of the 45 degrees seen in the 288 nm diameter case, Fig. 2. The MFM data shows strong contrast above every domain, indicating that none of the domains is closed on top, and thus are bubble type structures. The bull’s-eye structure is also found at this diameter. Sometimes the center domain is very small and remains in the particle center, while other times the eye shifts to one side. The MFM data clearly shows a difference in the contrast of the two domains indicating that one of them has more closure than the other. It is easy to imagine that the crescent pattern in Fig. 5 is actually a distorted bull’s-eye, and that such a distortion might be due to several causes. For example, it is possible that the field from the MFM probe distorts the pattern, e.g. distorting the bull’s-eye into the crescent shape similar to thatshown in Fig. 5. We note, however, that the simulated crescent patterns, such as that shown in Fig. 5 b,d,e, and f, have so far not been found to be even metastable; we have seen simulations evolve through a crescent, and we have also initialized the simulation with a crescent shape, but in all cases these evolve to a different final metastable structure. It is also possible that the LLG simulations do not (meta-) stabilize in a crescent structure because of the regularity of the model: We use identical grains on a perfect lattice in our simulations, while in nature this is unlikely to be the case. For example, deviations from perfect order may in some cases provide pinning sites for the magnetization.SINGLE DOMAIN PARTICLES (1/2 Magnetic Stripe Period and Smaller)The two-stripe structure is found to be stable in the simulations until the diameter is decreased to approximately 84 nm. Similarly, the experiments indicate that there is a structural change when the two stripe structure is decreased in diameter: Particles are often found that are not stable enough to be imaged by MFM. Streaking images across the particle indicate that the magnetization is suffering changes during imaging; this is quite prevalent experimentally around 140 nm in diameter. Around 125 nm in diameter several stable particles are found which exhibit little MFM contrast. As shown in Fig. 6, these must be closed structures and are likely to be in-plane vortex, two-stripe like structures with little emanating field. The perpendicular domains are almost completely closed except for a small region near the edge of the particle, indicating that little magnetic flux leaves the particle, making it almost invisible to the MFM probe.The competing state at this size is also found. When starting from perpendicular saturation, the simulations produce a flower state, or single domain state, an example of which is shown in Fig. 7.These two structures are not due to differences in the particles’ fine structure, as was demonstrated experimentally by saturating the same particles and re-imaging several times. Both states were found to be stable in each individual particle. The simulations produce similar results; when started from the two-stripe structure the vortex is stable, while the flower state is the stable state when started from saturation.SUMMARY AND CONCLUSIONSThe experimental occurrence of differing magnetic structures as a function of particle diameter is summarized in Fig. 8a. Since particles have been seen to assume more than one metastable state both experimentally and theoretically, the frequency of the appearance of different states at a given size was investigated. Experimentally imaging particles of different sizes in both the as-deposited and post-saturation states did this. The percentage of particles at a given size in a given state is shown in Fig. 8a.Although these states are presumably all metastable states, the value of their energy gives some insight into the occurrence of differing structures as a function of particle diameter. Fig. 8b shows the energy density of the states at different sizes. A state with lower energy density should be the favored state at a given diameter if there is a dynamical path to reach that state. As can be seen, the single domain is the only structure found below 80 nm. The in-plane vortex and single domain states both appear up to150 nm, followed by both the two-bubbles and bull’s-eyes to 200 nm. The three-stripe and bull’s-eyes structures are energetically most favorable above 250 nm, with an occasional two-bubble state appearing in this region.In conclusion, we see that geometrically regular nanoscale magnetic particles may be fabricated, and their full magnetic structure may be determined by a combination of high-resolution MFM and micromagnetic simulations. The resultant magnetic structure is a rich function of the particle size, ranging from single-domain flower structures at the smallest diameters, through multi-domain structures characterized by different in-plane vortex patterns, culminating in many complex striped states at diameters near and above 1 µm which resemble the stripe structure of magnetic films of the same 100 nm thickness. The implication of these results for practical applications awaits further exploration of such issues as the longevity of these metastable magnetic structures and the mechanisms of spontaneous magnetic structural change such as magnetic reversal.ACKNOWLEDGMENTSWe wish to acknowledge assistance or helpful discussions with Dr. Eric Granstrom, Dr. Jake Schmidt, and Dr. Christopher Merton, as well as partial researchsupport from the University of Minnesota Graduate School, and the University of Minnesota Supercomputing Institute for Digital Simulation and Advanced Technology. REFERENCES1. Campbell, S. A. “The Science and Engineering of Microelectronic Fabrication,”pp.206-214, (Oxford Press, New York, 1996).2. Boerner, E. D. and Bertram, H. N. “Non-Arrhenius Behavior in Single DomainParticles,” IEEE Trans. Mag., Vol. 34, pp.1678-80, 1998.3. Skidmore, G. D. and Dahlberg, E. D. “Improved Spatial Resolution in MFM,”Appl. Phys. Lett., Vol. 71, pp. 3293-95, 1997.4. G. Wastlbauer, G.D. Skidmore, et al., “MFM Studies of Magnetization Reversal inCoCr Thin Films,”5. Skidmore, G. D. Chap. 3, University of Minnesota Ph.D. Dissertation, 1998.6. Kunz, A., Skidmore, G. D., Campbell, C. E. and Dahlberg, E. D. to be published7. Skidmore, G. D. Chap. 2, University of Minnesota Ph.D. Dissertation, 1998.8. Digital Instruments, 112 Robin Hill Road, Santa Barbara, California USA 93119;9. Shi, J., Awschalom, D. D., Petroff, P.M. and Babcock, K. “Assembly and imagingof submicron ferromagnets in GaAs,” J. Appl. Phys. Vol. 81, pp. 4331-35, 1997.10. Boerner, E.D. and Bertram, H.N. “Dynamics of Thermally Activated Reversal InNonuniformly Magnetized Single Particles,” IEEE Trans. Mag., Vol 33, pp.3052-54, 1997.11. Brown Jr., W.F. “Micromagnetics,” (Wiley-Interscience, N.Y., 1963); Chikazumi,S. “Physics of Ferromagnetism,” (Oxford Press, London, 1997).12. Cullity, B.D., “Introduction to Magnetic Materials”, p. 234, (Addison-Wesley,Reading, MA, 1972).13. Scheinfein, M. and Price, E., “LLG Micromagnetics Simulator”, Tempe, AZ USA85282.FIGURE CAPTIONSFigure 1. Images of micron diameter nickel particles. Figure a shows gray scale experimental MFM images, and figure b shows a false color representation of simulated LLG particles with 900nm diameter. Both sets of pictures show complicated and uncharacterized stripe and bubble structures. In the gray scale, white corresponds an attractive interaction with the magnetic tip, and black is a negative interaction. Gray means no interaction. In the false color images, red corresponds to moments pointing out of the page, with blue into the page, and black in the plane of the page.Figure 2. Bull’s-eye magnetic structure of a 288 nm in diameter, 100 nm high nickel cylinder. The bull’s-eye structure is characterized by a circular domain wall closing around the particle. Images a and c are oppositely MFM images taken with the tip oppositely magnetized, while Fig. b shows the LLG simulated magnetization structure at the top of the particle. Figs d and e show the magnetization directions of the grains at the top circular surface and in the mid-plane slice parallel to the top of the cylinder, represented as cones. Fig. f is a slice vertically through the particle, again showing the magnetization direction with cones.Figure 3. A three-stripe structure in a 288 nm diameter particle. Fig. 3 differs from Fig.2 in that the circular domain wall does not close on itself, but turns out towards the edges. This is clearly seen in figure e, as well as in f, where the centers of the 2 walls point in the same direction, as opposed to Fig. 2f.Figure 4. A two-stripe domain pattern in a 200 nm diameter particle. Fig. f, the vertical slice through the particle shows a vortex like pattern separating the up and down domains. The high contrast shown in the MFM images a and c is due to the large about of magnetization pointing out at the surface of the particle.Figure 5. Crescent structure in a 200 nm diameter particle. The difference in contrast shown in the MFM images demonstrates differing amounts of closure in the up and down domains. We show this more clearly in the vertical slice, Fig. f, where the down domain iscapped, while the up domain points out at surface of the particle. The crescent pattern is not stable in the LLG simulations, suggesting that perhaps pinning sites exist the experimental systems, or possibly a distortion due to the probe.Figure 6. An in-plane vortex structure in a 150 nm diameter particle. The almost non-existent MFM pattern in figures a and c is the consequence of the almost full closure or vorticity though the thickness of the particle shown in figure d.Figure 7. A flower magnetization structure in a 150 nm diameter particle. Figs a and c clearly show differing attraction to the magnetic tip due to the opposite magnetization of the tip. Fig. d shows the magnetization along a vertical slice where the magnetization flowers out towards the sides of the particle.Figure 8. Fig. a shows a plot of the percent of times a given structure appears at a given diameter experimentally. Below 80 nm, only single domain flower patterns appear. As you increase in diameter, the in-plane vortex becomes a stable state. Increasing further in diameter, single domain behavior is wiped out, but crescents, two stripes, bull’s-eyes, and three stripe particles begin to appear. Fig. b shows the energy of the given structure at each diameter. The lowest energy particle is most favorable. The results of these energy calculations match well with what is seen experimentally.。