Work is going on in many labs around the world to find industrial processes to produce graphene and make use of it (you can produce graphene yourself by drawing a line with a pencil on a sheet of paper: it is very likely that somewhere in that line there are single atom layers that we called graphene). Others are studying the electrical properties of graphene...
This latter part is fascinating. So far scientists (and engineers) have been working with solids, 3D structures and have learnt the properties of the surfaces of these solids (as an example we understand very well that electrons in a conductor tend to move to the surface of the wire, and the higher the frequency the more this effect is important). With graphene, but also with new single atom layers of materials (like molybdenum), we no longer have a 3D structure but a 2D one and we need to understand what happens at the edges of this surface, that is in the 1D dimension.
Researchers at the University of Pennsylvania have used a state of the art aberration corrected electron transmission microscope to study the electrical properties of graphene nano ribbons, GNR, ribbons with a width of 10nm.
In the words of Marija Drndic: new text
“Graphene looks like chicken wire, and you can cut up this hexagonal lattice of carbon atoms in different ways, producing different shapes on the edge. But if you cut it one way, it might behave more like a metal, and, if you cut it another way, it could be more like a semiconductor.”
It is quite amazing to see how we are approaching the mathematical "line". Through a mathematician will disagree (we are talking about something that is 0.1 nm thick!) in engineering terms there is nothing below that thickness and from an engineering standpoint we have really reached the 1D line. Who could have thought, just 20 years ago, that we would have reached the point where the orientation of atoms on a surface would be visible and would make a difference?