Two-dimensional imaging of electronic wavefunctions in carbon nanotubes

The drive towards the development of molecular electronics is placing incre asing demands on the level of control that must be exerted on the electroni c structure of materials. Proposed device architectures ultimately rely on tuning the interactions between individual electronic states, which amounts to controlling the detailed spatial structure of the electronic wavefuncti ons in the constituent molecules(1,2). Few experimental tools are available to probe this spatial structure directly, and the shapes of molecular wave functions are usually only known from theoretical investigations. Here we p resent scanning tunnelling spectroscopy measurements of the two-dimensional structure of individual wavefunctions in metallic single-walled carbon nan otubes; these measurements reveal spatial patterns that can be directly und erstood from the electronic structure of a single graphite sheet, and which represent an elegant illustration of Bloch's theorem(3) at the level of in dividual wavefunctions. We also observe energy-dependent interference patte rns in the wavefunctions and exploit these to directly measure the linear e lectronic dispersion relation of the metallic single-walled carbon nanotube .