70’s-Era Physics Prediction Finally Confirmed

Artist’s illustration of a butterfly as if departing from a moiré pattern in graphene formed on top of a sheet of boron nitride. (Credit: James Hedberg)

New CCNY professor part of team confirming Hofstadter Butterfly in graphene

City College of New York Assistant Professor of Physics Cory Dean, who recently arrived from Columbia University where he was a post-doctoral researcher, and research teams from Columbia and three other institutions have definitively proven the existence of an effect known as Hofstadter’s Butterfly.

The phenomenon, a complex pattern of the energy states of electrons that resembles a butterfly, has appeared in physics textbooks as a theoretical concept of quantum mechanics for nearly 40 years. However, it had never been directly observed until now. Confirming its existence may open the door for researchers to uncover completely unknown electrical properties of materials.

“We are now standing at the edge of an entirely new frontier in terms of exploring properties of a system that have never before been realized,” said Professor Dean, who developed the material that allowed the observation. “The ability to generate this effect could possibly be exploited to design new electronic and optoelectronic devices.”

The international group, which also included the University of Central Florida, the National High Magnetic Field Laboratory, and Japan’s Tohoku University and National Institute for Materials Science, published its findings in the journal Nature; they appeared in an advance online publication May 15. Separate groups at the University of Manchester (UK) and Massachusetts Institute of Technology simultaneously reported similar results.

Douglas Hofstadter, a physicist and Pulitzer Prize-winning author, first predicted the existence of the butterfly in 1976, when he imagined what would happen to electrons subjected to two forces simultaneously: a magnetic field and the periodic electric field.

The energy spectrum, or pattern of energy levels, that these dueling forces create is said to be “fractal,” that is, infinitely smaller versions of the pattern appear within the main one. This effect is common in classical physics, but rare in the quantum world.

“When you plot the spectrum, it takes on the form of a butterfly. Zoom in on the spectrum and you see the butterfly again, zoom in and see butterfly again,” said Professor Dean. The light and dark sections of the pattern, respectively, correspond to light “gaps” in energy level that electrons cannot cross and dark areas where they can move freely.

“The existence of gaps changes the way electrons move through a material. Copper for example, has no gaps, whereas an insulator, like glass, has very large gaps,” explained Professor Dean. “The relationship between energy and how dense the electrons are in a material – energy density – determines all electrical properties. That’s why copper conducts, glass or ceramic doesn’t, and other materials weakly conduct, like semiconductors.”

“What you see in a Hofstadter spectrum is a very complicated structure of gaps arranged in a fractal pattern,” he continued, which suggests as yet unknown electrical properties.

The team produced the effect by sandwiching together flat sheets of graphene – a single-atom-thickness of carbon – and another material, called boron nitride, and twisting them against each other to create what is called a superlattice. “Graphene has hexagonal chicken wire structure and boron nitride does too,” he said. “It is as if you take screen door material and put one sheet on top of other. As you rotate it you see a periodic pattern appear. You get an interference effect – a ‘moiré’ pattern.” In the case of the chicken-wire structure of graphene and boron nitride, the pattern forms a fractal butterfly of energy states.

“This is a very good example of fundamental discovery that opens doors that we don’t even know about yet. Why go to a distant planet?” Professor Dean wondered, about the implications of the work. “We go there to discover what’s out there. We don’t yet know what this new world will result in and what will emerge out of this.”


  • C. R. Dean, Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices, Nature, May 15, 2013 doi:10.1038/nature12186

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