The key to making useful nanoelectronic devices from graphene is to first understand, and then be able to control, the flow of electrons through tiny snippets of the material. The absence of a bandgap in pure graphene means that although its electrical conductivity is the highest of any material bar none, it is nearly impossible to shut it off completely. Researchers at MIT and elsewhere have recently figured out not only how to build precisely defined bandgaps into composites of graphene and boron nitride, but they have also uncovered the deeper electronic structure of the material and found that it contains some of the most fascinating physics known.

What the MIT researchers basically did was take single layers of hexagonal graphene and stack them up against single layers of hexagonal boron nitride. The key is to be able to control the degree of alignment between the layers, and therefore the ease with which electrons can hop and slide from one layer to the next. In order to coax the graphene-boron honeycomb into exposing its hidden behaviors, some additional outside influence needs to be imposed. One effective way to do this is to chill everything down to within a fraction of absolute zero and add a massive, out-of-plane magnetic field.

When you do that, it becomes possible to see and measure a rarefied theoretical beast known as ‘Hofstadler’s Butterfly’. This fractal pattern emerges when the electronic energy levels of the material are plotted against the applied magnetic field. Originally proposed by Douglass Hofstadler in 1976, his signature butterfly has only recently been experimentally observed in the lab. Another way to state more precisely what these ‘energy levels’ really are is to define them as the observable property of the wave function, or more specifically, as the plotted electron density. In the present case, the researchers used fields up to 45 Tesla that were available at the National High Magnetic Field Laboratory in Tallahassee. For comparison, that’s about five times as large as the most powerful MRI machine in common use.

As a graphical representation of the fractal structure of the energy spectrum for electrons in a magnetic field, the butterfly structure has an intrinsic ‘self-similarity’ which can be quantified by a parameter known as the fractal dimension. Intriguingly, just a short time ago, other researchers have found evidence for this exact kind of ‘quantum critical’ semiconductor behavior in many proteins commonly used for various enzymatic functions in cells. They were even able to calculate this fractal dimension for several of them. We won’t delve too much further into how it is figured (there are different ways), other than to say that it is a general measure of complexity we can casually define for our purposes here as the ratio of the change in detail to the change in scale.

When the graphene-boron honeycombs are stacked out of alignment, they create something known as a ‘Moire pattern’. When the angle of rotation between the layers was low, the material showed an insulating behavior, and when high, the conducting state was seen. Butterflies and Moire patterns aside, what really has the researchers excited is some of the other physical effects they were able get from graphene. The researchers previously demonstrated something known as a ‘quantum spin Hall state’ when they applied a magnetic field with an in-plane orientation. The field forced electrons at the edge of the material to move in opposite directions, and in separate lanes, according their spin. In contrast to the unidirectional current flow of electrons in a regular metal, a material that behaves as a ‘topological insulator’ would be useful in several spintronic applications.

If all that terminology isn’t enough physics for you, there’s more. While the famous Schrödinger equation (which gives the wave functions mentioned above) describes the behavior of electrons in most materials, electron behavior in graphene is ‘ultrarelativistic’ and therefore is better described using the lesser-known Dirac equation. Compared with normal materials, where electron velocity is subrelativistic, electrons in graphene composites configured with just the right alignment can flow at significantly greater speeds, and need to be described with a different formalism. Furthermore, when many layers of graphene are properly stacked together (with associated greater strength), they can still show the high conduction seen in a single layer.

There are still many details to be worked out before we can put graphene-based semiconductor devices into mainstream use. However, where before we couldn’t say much more than that electron flow within a layer is expected to be significantly different from that across layers, we can now point to a much richer physics of graphene.