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he magnetic structure of thin magnetic films has long been a subject of importance for information storage media, sensors, and technological advances such as magnetic RAM. In many applications, and for modeling and simulations, films are composed of small magnetic particles that interact with each other through dipolar and exchange coupling. These films can be pictured as an array of tiny magnets whose magnetization can point in any direction in three dimensions.
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| Figure 1: A model of a 1 X 2 micron permalloy film in differing magnetic states. Saturation in the easy action direction (left), final state after relaxation in zero applied magnetic field at a temperature of 300 K (middle), and as in the middle but at temperature 100 K (right).
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Experimentally, the magnetic structure is studied by magnetic force microscopy (MFM), in which a tiny magnetized stylus is brought into proximity to the surface of the film and then rastered over the film. The magnetic force between the film and the stylus is then interpreted to elucidate the magnetic structure. Physics Professor E. Dan Dahlberg, Director of the Magnetic Force Microscopy Center (MMC) at the University of Minnesota, has helped develop and employ sensitive MFM to study these films and magnetic particles.
Useful insight and interpretation of these measurements are obtained by modeling these systems and simulating their magnetic structure and the evolution of that structure as a function of time, temperature, and applied magnetic field. Physics Professor, Charles E. Campbell, an Associate Fellow of the Supercomputing Institute and member of the MMC, graduate student Andrew B. Kunz, also a member of the MMC, and Professor Dahlberg are developing and implementing simulation methods to study these systems. They make use of both micromagnetic methods, wherein the dynamical Landau-Lifshitz-Gilbert equations (LLG) are solved at zero temperature, and Monte Carlo simulation techniques with a Metropolis algorithm to study both the Monte Carlo time and temperature dependence of the relaxation. Monte Carlo simulations, approximated by the evolution of the system as a function of Monte Carlo steps, show that the difference between room temperature and zero temperature is very important for some systems being simulated.
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| Figure 2: Total magnetization as a function of applied field for the film of figure 1, beginning in a saturated state at applied field +600 Oe (as in Figure 1 left), and then stepping the field down to 600 Oe in steps of 10 Oe(top curve) at reverse saturation, and finally returning the applied field of +600 Oe near the original saturated state (bottom curve).
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One set of simulations is for a model of micron size permalloy films with a
thickness of 20 nm that consists of a two-dimensional square mesh of grains.
Each grain occupies a size of 50 X 50 X 20 nm and has an associated magnetic
moment allowed to rotate fully in three dimensions. The permalloy is characterized
by a uniaxial anisotropy situated along one of the principal axes of the film
of 5000 erg/cc&emdash;a nearest neighbor exchange of 1.3 10-6 erg/cm, and a saturation magnetization of 800 emu/cc. Figure 1 (left) shows a 1 X 2 micron film whose magnetization was initially saturated in the long direction of the film (taken to be the easy axis) and permitted to relax in the absence of an external magnetic field. The final state at room temperature, figure 1 (middle), has a complex magnetic structure in which the magnetizations at the ends have rotated to parallel the ends in order to reduce the demagnetization energy, and vorticity has developed in the interior to lower the dipolar and exchange energies. Even though there is no applied magnetic field, there is a residual net magnetization in the direction of the original saturation. The outer edges have preserved the original direction, while the interior has a significant region of reversed magnetization. This is a metastable state; there is at least one configuration with lower energy which has higher symmetry, but which is unlikely to be reached in an experimental relaxation. Figure 1 (right) shows the results of the simulation at 100K, demonstrating an apparent strong dependence on temperature.
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| Figure 3: Magnetic states at the applied fields of 60 Oe (left) and 70 Oe (right) on the demagnetization curve (top curve in Figure 2).
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Simulations of the same system have been carried out as a function of magnetic field. Figure 2 is a hysteresis graph. Beginning at a high positive field along the long axis, the initial state is the same as Figure 1 (left). The field is first reduced to 600 Oe, and then in steps of 10 Oe until -600 Oe, where the magnetization is well-saturated in the opposite direction of the initial state. The magnetization as a function of applied field during this reversal is the top curve. At -600 Oe, the field is increased in steps of 10 until it reaches 600. The lower curve is the magnetization as a function of field as the field is increased from 600 Oe to +600 Oe. The large discontinuities in the hysteresis curves are a real physical effect,
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| Figure. 4: Magnetic structure of the top of Ni cylinders. An array of cones in the direction of magnetization is shown.
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due to a sudden move from one metastable configuration to a significantly different configuration during a very small change in the field. One such shift in magnetic structure change is shown in Figure 3 for the applied magnetic fields of -60 Oe to -70 Oe.
The magnetic structure of mesoscopic sized particles is a subject of growing interest at the fundamental level. MFM measurements on nickel cylinders of 100 to 300 nm diameters and 100 nm high indicate that the smaller particles are single domain magnetic structures while larger diameter particles have a complex magnetic structure. Andrew Kunz has simulated such particles using the micromagnetic LLG method, obtaining semiquantitative agreement with the MFM results. Figure 4 shows the magnetic structure of the top surface of a 200 nm cylinder where the initial magnetic configuration was uniform saturation parallel to the cylinder's axis. This is shown through false color images showing the direction of the x, y, and z components of the top surface, where y is the direction perpendicular to the page. The middle panel, showing the y component, is very similar to the MFM image. Since this is an LLG simulation, it is at a temperature of absolute zero.
The incorporation of finite temperature effects in the LLG method is a high priority challenge for the future of these simulations. These methods will also be applied to the structure and stability of magnetic bits on magnetic thin films and the more esoteric problem of spontaneous reversal of magnetism via quantum mechanical tunneling.
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URL: http://www.msi.umn.edu/general:80/Bulletin/Vol.16-No.1/5.html
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