Fractals add new dimension to study of tiny electronics

When it comes to miniature electronics, scientists have seen the shape of things to come — and that shape is a fractal.

People most often see fractals in the familiar, irregular branching shapes of nature — a leaf, or tree, or snowflake. A repeating pattern of ever-smaller branches gives these structures a unique profile that defies classical geometry.

Now a study suggests that magnetic fields can take the form of fractals, too — if a magnet is made of plastic molecules that are stacked in parallel chains.

While the results could influence the design of electronic devices in the distant future, the work is so new that scientists are only beginning to consider its implications, said Arthur Epstein, Distinguished University Professor of chemistry and physics and director of the Center for Materials Research at Ohio State University.

Epstein and longtime collaborator Joel Miller, professor of chemistry at the University of Utah, described the study in a recent issue of the journal Physical Review Letters. Coauthors included graduate students Stephen Etzkorn at Ohio State and Wendy Hibbs at the University of Utah.

Using a computer model, the scientists tried to look ahead to a time when electronic structures can be built so small that they no longer behave like normal three-dimensional objects.

“The materials currently used in magnetic devices — for example, computer hard discs or ID strips on credit cards — behave like three-dimensional magnets,” explained Epstein. “However, the decreasing size of these devices may one day require them to be considered one- or two-dimensional in nature. As the spatial dimensions decrease, the magnetic dimensions of the materials may take on fractal values.”

Mathematically, fractals are considered to exist in partial, or fractional, dimensions. That means if a device produced a magnetic field that exhibits fractal behavior, the magnetic field wouldn’t possess dimension equal to a whole number — such as one, two, or three dimensions — but rather a fractional value such as 0.8 or 1.6 dimensions.

Such a seemingly bizarre existence in fractional dimensions sounds like the stuff of science fiction, but that’s what Epstein and his colleagues found when they modeled the behavior of a plastic magnet.

The model consisted of a hybrid material, a compound of manganese tetrapheynlporphyrin and tetracyanoethylene. Theoretically, this compound can form polymer chains that are one-dimensional.

The researchers modeled the behavior of the material as it was magnetized by an external magnetic field and then cooled to a critical temperature where it began to behave as a special kind of glass. At -267ºC (-449ºF), the magnetic field of the material appeared to exist in 0.8 dimensions. As it cooled a little further, it gradually became one-dimensional, then finally settled at 1.6 dimensions at -269ºC (-452ºF).

The “spin” of the molecules — a quality that relates to the source of magnetism and magnetic fields in materials — appeared to form clusters within the material, with each cluster pointing its magnetic field in a different direction. Many magnetic fields sprouted out from the material like branches of a cactus. Tiny secondary magnetic fields then sprang out from the branches like needles on a cactus.

Eventually, the cacti-like branches were locked together, with crisscrossing needles holding them in place. This interlocking fractal growth gave the magnetic field a unique kind of order, and as a result, the material would be called a “fractal cluster glass,” Epstein said.

To explain this behavior, Epstein likened the one-dimensional polymer chains of this exotic compound to stacks of poker chips. “Imagine each poker chip is an atom,” he said, “and that many stacks of chips are immersed in chicken fat.”

“When the fat insulation is thin, the stacks of poker chips can all ‘see’ each other. When it comes to orienting their magnetic fields, each stack can look to its neighbor to see what it should be doing, and all the stacks can orient the same way,” he said.

“But when the chicken fat insulation is thick (as for the materials used in this study), it becomes opaque, and suddenly the stacks of poker chips can’t see each other,” he continued. “Without knowing what its neighbors are doing, each stack has to pick a random direction for its magnetic field.”

Spins of different groups of atoms within the stack may pick different directions at first. Normally, these clusters would interact with each other in a kind of competition, until the magnetic fields were all pointing in the same direction.

“Sometimes the atoms don’t have enough energy to fight with each other. Then the clusters just continue to point in random directions, and the material is considered a cluster glass,” Epstein said.

Epstein believes that cluster glasses could play a role in future electronics, when organic-based magnets could be structurally tuned to provide magnets of differing dimension.

Until then, scientists will have to probe this fractal behavior further.

“We’re just now beginning to understand these exciting phenomena,” Epstein said.

The Department of Energy and the National Science Foundation supported this work.

Arthur J. Epstein, (614) 292-1133; Epstein.2@osu.edu

Written by Pam Frost Gorder, (614) 292-9475; Gorder.1@osu.edu