Per Arne Rikvold

As computers become ever smaller and more powerful, the need for ultrahigh-density data recording media is increasing. One way to achieve this is to record single bits of information onto individual, nanometer-sized magnetic particles. However, information stored on such small particles becomes sensitive to weak magnetic fields and thermal fluctuations. In order to create high-density recording media which retain their information at room temperature in reasonable magnetic fields (such as the earth's magnetic field or airport metal detectors), it is necessary to understand how the magnetization of such nanoparticles depends on the particle size, the applied magnetic field, and the temperature.

In our research group, we have studied magnetization switching in nanometer-sized ferromagnets at nonzero temperature for several years. There are still significant uninvestigated issues about this topic, which it is obviously important to settle since most recording devices operate at room temperature.

Fig. 1: A metastable free-energy well is schematically illustrated in (a). Although the stable state is on the right, a free-energy barrier F must be overcome in order for the metastable state on the left do decay to the stable state. As the applied magnetic field is increased, the barrier decreases. It becomes zero at a particular field called the nucleation field, as illustrated by the heavy curve in (b). For stronger fields there is no metastable state, as illustrated by light curve in (b).

We actually started this work because of our interest in a fundamental problem in nonequilibrium statistical mechanics: metastability. As an example, let us consider a compass needle. If it is free to rotate, it will point to the north-parallel to the earth's magnetic field. That is its stable position. If, instead, the needle is fixed so that it points to the south, it is in a metastable state. If the south-pointing needle cannot rotate, like the magnetic particles that are glued to the surface of a recording tape or disk, the magnetization of the needle itself has to reverse if it is to point in the stable direction. This is what we call magnetization switching. In order for the magnetization to reverse its direction, the microscopic magnetic moments that make up the magnet have to rotate across a free-energy barrier consisting of states that have much higher energies than either the metastable or the stable state. Such a free-energy barrier is schematically illustrated in Fig. 1. The energy required to do this must be "borrowed'' from the surroundings as thermal fluctuations. If the barrier is high or the temperature low, the average lifetime of the metastable phase can therefore be extremely long.

In fact, magnetic particles are only one example of metastablity in nature. Perhaps the most familiar, at least to those of us who grew up in cold climates, is freezing rain. Pure water can be supercooled far below its freezing point, and if it is not disturbed, it can remain fluid for a very long time before it freezes. So supercooled raindrops stay liquid until they hit something hard, when they turn into a dangerous and beautiful layer of clear ice, covering roads, wires, trees, and buildings. The first scientific description of a metastable phase that I am aware of, is a paper on supercooled water by Fahrenheit in the Philosophical Transactions of the Royal Society of London from 1724. Other common examples of metastable phases include ferroelectric materials, diamond-a metastable form of carbon, and vortex states in superconductors. Outside condensed-matter physics we have, for example, the "false vacuum" associated with the electroweak transition, and the supercooled quark/gluon plasma associated with the QCD confinement transition.

It is the very long time that "nothing" happens in a metastable system which makes computational studies of magnetization switching and other examples of metastable decay so challenging. If you try to simulate the time evolution of a particle which is magnetized in a direction opposite to the magnetic field, most of the time you only see microscopic fluctuations near the metastable state. But once in a very long while, many such tiny fluctuations come together to create one which is large enough to take the particle across the free-energy barrier and into the stable state. The only problem is that the microscopic fluctuations only last for less than a billionth of a second each, while the large fluctuation needed to get out of the metastable state may only occur once in a few years, decades, or more! Even if you could simulate the microscopic fluctuations at the actual rate that they occur (which mostly you cannot), it might take years to simulate a single switching event, and centuries to get enough statistics to publish your results. The jargon for this problem in computational materials science is "Bridging the Time Scales,'' and it is currently something of a Holy Grail for researchers simulating dynamical materials properties.

Our group has developed several advanced algorithms that go a long way towards solving this probelm, at least for some simple systems. These include Dr. Novotny's Monte Carlo with Absorbing Markov Chains method; a method we call Projective Dynamics, which we developed with our postdoc, Miroslav Kolesik; and a transfer-matrix method which we developed with our former graduate students, Christoph Günter and  Bryan Gorman. These methods can give speedups of more than twenty orders of magnitude, compared to "naïve" Monte Carlo simulations, without changing the underlying dynamics! Some of the algorithms have been parallelized on a Cray T3E by our postdoc, György Korniss.

We have used these algorithms to simulate switching in models of highly anisotropic magnetic nanoparticles and ultrathin films. We did this work together with our former graduate students, Howard Richards and Xuekun Kou, undergraduate Dean Townsley, and postdoc Raphael Ramos, and our present graduate student, Steven Mitchell, as well as collaborators in the US and abroad. These studies showed that the metastable lifetime depends on the particle size, as well as on the temperature and the strength of the applied field, in ways that we are able to understand using nucleation theory and the Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory of phase-transformation kinetics.

Hysteresis is an important aspect of the behavior of any magnetic material subject to a time-varying field. It has been an active area of research for over a century, ever since the work by Ewing, Warburg, and Steinmetz in the 1880s, which was motivated by the need to understand energy loss and heating in the high-tech devices of their time: electrical motors and transformers. Today we are asking the same questions about magnetic nanoparticles and ultrathin films, and we have used our advanced algorithms to study hysteresis in dynamic spin models. This work was the core of Scott Sides' Ph.D. dissertation in 1998, and it is now being continued by György Korniss and undergraduate Christina White. One phenomenon we are excited about is a novel dynamic phase transition which occurs as the frequency of the oscillating field is changed.

Fig. 2: Snapshots of the z-component of the magnetization for a 9 nm x 9nm x 150 nm iron nanomagnet during a simulated switching event (shown in 3/4 cut-away view). Blue indicates magnetization parallel to the applied field, while red indicates antiparallel (metastable) magnetization. Initially, the bulk magnetization is antiparallel to the field, but the magnetostatic interactions cause end caps where the magnetization is weaker (green). The magnetization reversal shown here occurs at a temperature of 20 K and a field of H = -1850 Oe. It is initiated by thermal fluctuations in the lower cap, which afterwards grows to switch the bulk magnetization. This simulation took about 50 hours on a DEC-Alpha workstation, using Dr. Brown's finite-temperature micromagnetics code. Several hundred runs are needed to obtain satisfactory statistics. Further details and movies available at http://www.csit.fsu.edu/~browngrg/micromag/

The simple models we have studied in the past are adequate representations of only a few real magnetic systems. To study more realistic models, our postdoc, Gregory Brown, has recently developed a finite-temperature micromagnetics code, using a stochastic differential-equation technique. We are now using this code to simulate magnetization switching in nanometer-sized iron pillars. Figure 2 shows a series of snapshots of a simulated switching event in a 9 nm x 9 nm x 150 nm iron pillar at a temperature of 20 K. We are particularly excited about this project because such pillars are actually manufactured in our department by MARTECH Director Stephan von Molnár's group, and we plan to concentrate much of our work in this area in the future.

In summary, computational and theoretical studies of magnetization switching dynamics in nanoparticles and ultrathin films is an exciting research area in which basic science and applied research can progress in parallel, each enhanced and stimulated by the other.

I am happy to thank the sponsors of our research: the National Science Foundation and the Department of Energy, as well as FSU through MARTECH, SCRI, the Department of Physics, and the new School for Computational Science and Information Technology (CSIT). Movies and more information about our group are found at http://www.csit.fsu.edu/~rikvold/matsci_html/matsci-mag.html, and my e-mail address is rikvold@csit.fsu.edu.

Some light reading

1. P. A. Rikvold and B. M. Gorman, "Recent Results on the Decay of Metastable Phases,'' in Annual Reviews of Computational Physics, edited by D. Stauffer (World Scientific, Singapore, 1994), pp. 149-191.

2. P. A. Rikvold, M. A. Novotny, M. Kolesik, and H. L. Richards, "Nucleation Theory of Magnetization Switching in Nanoscale Ferromagnets,'' in Dynamical Properties of Unconventional Magnetic Systems, edited by A. T. Skjeltorp and D. Sherrington (Kluwer, Dordrecht, 1998), pp. 307-316.

3. M. A. Novotny and P. A. Rikvold, "Magnetic Particles,'' in Encyclopedia of Electrical and Electronics Engineering, Vol. 12, edited by J. G. Webster (Wiley, New York, 1999), pp. 64-73.