Megan Shumaker(a,b) and Dr. Selman Hershfield(b)
(a)Kansas State University, Manhattan, KS
(b)Department of Physics, University of Florida, Gainesville, FL
Micromagnetic modeling uses computers to predict the magnetic behavior of materials on short length scales of the order of nanometers. It is an interesting field right now, especially due to faster computers, new microscopies, and a desire for improvements in magnetic media. A brief overview of two publicly available micromagnetic simulators is presented.
The basic concepts of magnetism have been well understood for many years. Magnets are used throughout industry, in everything from children’s toys to motors to hard disk drives. As technology continues to advance, we find we need to understand magnetism on a much smaller-length scale. At lengths under a micron, macroscopic magnetic models are no longer adequate to describe the behavior of the magnetic material. This is where micromagnetics comes in. The study of micromagnetics strives to understand magnetism on the smallest-length scale possible.
To start out, a basic introduction to magnetism is necessary. There are three types of magnetic materials. Diamagnetic materials, such as bismuth and antimony, are weakly repelled by both poles of a magnet, so they become magnetized opposite in direction to the external magnetic field . Paramagnetic materials, such as aluminum and platinum, react only slightly to a magnetizing force, and tend to magnetize in the direction of the external field.1 Ferromagnetic materials have a non-vanishing magnetization even in the absence of an external field. Ferromagnetism is generally what is meant when people talk about magnetism and will be the subject of the rest of the paper. Ferromagnetic elements include iron, nickel, and cobalt at room temperature, gadolinium below 16 degrees Fahrenheit, dysprosium at cryogenic temperatures, and a wide range of alloys.2 Ferromagnetic materials become magnetized when there are more electrons with their spins (and magnetic moments) in one direction than the other. Magnetic moments want to line up with the external magnetic field, because this is their lowest energy state. However, as you look at the field lines of a magnetic dipole (figure 1), it is easy to see that at the dipole, the field lines point in one direction, but a little way away, the field lines point in the opposite direction. This change in direction of the field lines is linked to an effect called the demagnetizing field, which ultimately causes the material to be divided up into small sections called domains. Within these domains, the moments tend to line up almost perfectly, but it takes a large force to align the net moments of each domain with each other. When they are aligned to the point where adding more energy would not align them any more, the material is at its maximum magnetization, or saturation magnetization. The magnetization process is not reversible, though, as shown by a hysteresis loop (figure 2).3 For instance, when an external magnetic field is applied, and then reduced back to zero, there is a residual magnetization left on the material. In other words, the moments tend to remain aligned to a certain extent. This is known as remanence. The coercive force is the opposing magnetic intensity required to remove the residual magnetism. Saturation magnetization, remanence, and coercivity are three properties of magnetic materials that are important when working on the microscopic level as well as the macroscopic level.
This brings the discussion to micromagnetism. As stated earlier, individual magnetic moments will want to align with an external field, until the local energy minimum is reached. If the external field changes, the moments will rotate and the domain patterns change and move to a new configuration, corresponding to the new local energy minimum.4 The main focus of micromagnetism is to study the magnetization reversal process and the complex domain configurations, especially in thin films. Thin films are important because of their use in the magnetic recording industry in data storage devices such as magnetic disks and magnetic tapes. To continue to fit more data into smaller spaces, it is essential to continue studying magnetism on smaller scales.
In recent years, micromagnetism has gained the interest of many researchers in both industry and academia. Micromagnetic modeling, used to simulate the magnetization process and predict complex magnetic domain structures, is experiencing growth because computers are rapidly becoming faster. At the same time, new experimental imaging procedures, with resolutions on the order of tens of nanometers have been developed to test theories. Electron holography, one form of imaging, produces an optical hologram with a resolution of about 1 nanometer (nm). Others, like scanning electron microscopy with polarization analysis (SEMPA), magnetic near-field microscopy, and magnetic force microscopy (MFM) have resolutions on the order of tens of nanometers. These two areas, imaging and modeling, complement each other, and data obtained from imaging can be used to test the models’ predictive capabilities.4
TWO MICROMAGNETIC SIMULATORS
A micromagnetic simulator is a computer program designed to compute the local magnetization within a material. The user provides inputs such as the material geometry, the initial magnetization, and the time evolution of the external magnetic field. It is also necessary to specify for each different material several parameters such as demagnetization, the exchange interaction field, and anisotropy. The exchange interaction field is a field between neighboring electron spins, which wants to align them to each other. Anisotropy represents a preferred direction of magnetization, which is usually in the direction of one or more of the major axes of the crystal lattice. In a material that has high uniaxial (along one axis) anisotropy, the domain structure may be as shown in figure 3.
We have tested two software packages, developed to model micromagnetism; the Object Oriented Micro Magnetic Framework (OOMMF) and the PC Micromagnetic Simulator (Simulmag). The first program, OOMMF, is a project of the Mathematical and Computational Sciences Division of the Information Technology Laboratory (ITL) at the National Institute of Standards and Technology (NIST), developed mainly by Mike Donahue and Don Porter, and distributed freely on the Internet.5 The main purpose of this project is to have a user-friendly micromagnetic package, which is flexible enough that code for it can be written and shared between users. It can be used on a wide range of UNIX platforms, Windows NT, and Windows ’95. OOMMF will display the magnetic moments of a 3-dimensional object in a 2-dimensional vector field, and calculate and display the changes in the material as an external magnetic field is applied, then changed in a series of steps.5
To understand how OOMMF works, it is important to review the basic framework of the program. When Oommf is first started, only the main window, mmLaunch, is opened. MMLaunch controls six other program strings of the program: mmProbEd, mmSolve2D, mmDisp, mmGraph, mmDataTable, and mmArchive.
The first step is to define a problem in the problem editor, mmProbEd. Defining a problem involves defining material parameters, as well as the part geometry, initial magnetization, and the applied field and how it will be changed. The initial magnetization can be chosen from among a large number of domain configurations. Four of these are illustrated in figure 4.
MMSolve2D, the solver, does all the calculations and outputs them to the remaining program strings: mmDisp, mmGraph, mmDataTable, and mmArchive. Within the solver are user interfaces to run and pause the simulation.
MMDisp displays a 2-dimensional picture of the magnetic moments and shows them as they rotate. This makes it easy to see when objects such as vortices form. In mmGraph, the user chooses from a long list of properties to graph. MMDataTable contains a long list of many of the same properties as mmGraph to be displayed in the form of a data table. Finally, mmArchive saves the data from a simulation to a file. These files contain a six-column matrix containing the x, y, and z-components of magnetization, and the x, y, and z-coordinates of all the magnetic moments calculated. All of these functions of OOMMF require that the user define how often to output data to them. A more complete User’s Guide for OOMMF is available at: http://math.nist.gov/oommf/doc/ .
The other micromagnetic simulator (Simulmag) is based on a PC. It is also available for free on the Internet.6 Simulmag is different from OOMMF in that groups of magnetic materials and circuits can be defined, then moved relative to each other. The output of Simulmag is a 3-dimensional movie of the evolution of magnetic moments, as well as plots of the magnetization and magnetoresistance of the elements.6
An advantage of Simulmag is that individual elements can be defined differently and put together in a group, to model, for instance, an inhomogeneous magnetic device. One example of Simulmag’s use would be to define two groups representing an advanced recording head and the magnetic structure in media. Then these two groups could be moved relative to each other to simulate the recording process.6 The simulation can be saved, then played back later. A user’s manual can be downloaded from http://math.nist.gov/oommf/contrib/simulmag/doc/smmanual.html .
Unfortunately, the ten-week REU program did not allow enough time to do a research project using the two programs, but several tasks were accomplished. Files from OOMMF were successfully downloaded into MATLAB, and MPEG movies made of the time evolution of the vector fields. Six frames of a movie are displayed in figure 5, showing vortex formation during a magnetization reversal process in a fictional 50 x 250-nm thin film. Downloading this data into MATLAB will allow calculation of other quantities, such as resistance, which cannot be computed directly within the micromagnetic simulator. Such calculations are being performed by Dr. Selman Hershfield and Tat-Sang Choy of the Condensed Matter Theory Group in the University of Florida Physics Department. They are working to write code that will calculate the resistance of the material as a function of the changing magnetization.
As micromagnetic modeling techniques are developed and tested, and their results are compared with experimental measurements, our understanding of complex domain configurations and magnetization reversal processes continues to improve. This knowledge will undoubtedly be exploited to its fullest by industry, with the large demand for improvements in data storage and video and audio recording.
The REU program and it’s organizers, Dr. Kevin Ingersent and Dr. Allen Dorsey, are greatly appreciated.
Tat-Sang Choy and Marshal Ward are acknowledged for their help with the computers.
Financial support by the National Science Foundation for this project is gratefully acknowledged.
FIG. 1. The magnetic field in the region around a magnetic dipole.
FIG. 2. A hysteresis loop showing the remanence, coercivity, and saturation magnetization of a ferromagnetic material.
FIG. 3. The magnetic domains of a material with high uniaxial anisotropy in the vertical direction.
FIG. 4. Four examples of magnetic structures available in the Oommf package: (a) four domains (b) seven domains (c) a vortex (d) an exvortex.
FIG. 5. Six frames of a movie of a custom-defined 50 x 250 nm thin film, produced by the Oommf package and recreated in MATLAB, showing vortex formation in a magnetic reversal process.