Structural Properties of Graphene and Carbon Nanotubes

Graphene was discovered in 2004 and has since sparked much interest in the field of condensed matter physics. Graphene is an atomically thin sheet of carbon arranged in a two dimensional honeycomb crystal. The MerminWagner Theorem predicts that a perfect crystal can not exist in two dimensional space, so it was surprising when graphene was first observed[1]. The existence of graphene has since been explained by the idea that graphene has an intrinsic roughness. This rippling makes graphene a nearly perfect two dimensional crystal in three dimensional space[2, 3], which is not forbidden. Graphene has been called the “mother of all graphitic forms”[4] because it can be wrapped into buckyballs, rolled into carbon nanotubes and stacked into graphite. High resolution images of a sheet of graphene and a multiwalled carbon nanotube are shown in Fig. 1 and Fig. 2. These materials are not only important new testing grounds for fundamental physics such as relativistic quantum mechanics and low dimensional thermodynamics, but also have potential applications to nanoscale technology such as high speed transistors and lasers. We can not investigate the structural properties of these graphitic materials using conventional optical microscopes because it is not possible to resolve anything smaller than a wavelength of the light used to illuminate the sample. Light in the visible spectrum has a wavelength λ ≈ 500 nm. Due to the sub-nanometer length scales of the structures of graphene and carbon nanotubes it is necessary to use an electron microscope to investigate their structural properties. We operate using electrons at 80 kV because graphene and carbon nanotubes are unstable under higher energy electron beams and samples are rapidly destroyed[5, 6]. An electron with this energy is traveling at almost one half the speed of light and we must take relativistic effects into account. The relativistic wavelength of an electron is,