Hofstadter's Butterfly, 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 but had never been directly observed - until now. 


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.

The team produced the effect by sandwiching together flat sheets of graphene – a single-atom-thickness of carbon – and another material, 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,"
said City College of New York Assistant Professor of Physics Cory Dean, who developed the material that allowed the observation.
"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. 

"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 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.


Published in Nature