The Hofstadter Butterfly and IR Spectroscopy of High-Mobility Graphene

Graphene is an atomically-thin sheet of carbon atoms with a hexagonal lattice struc- ture. The material’s remarkable electronic properties make it ideal for testing the physics of mescoscopic systems in two dimensions. Known as a zero band-gap semiconductor, graphene displays a characteristic linear dispersion relation at low energies. Two sheets of graphene stacked together, referred to as bilayer graphene, exhibits a second energy degeneracy as a unique hyperbolic cone in its dispersion relation. These electronic char- acteristics produce quantum effects that differ dramatically from their three-dimensional counterparts. In this thesis we present the foundational theory behind low-dimensional semiconductors and graphene as well as preliminary results of research on high-mobility bilayer graphene. The initial goal for this research was to directly measure Landau level transitions using Fourier Transform Infrared (FTIR) spectroscopy. While the device was too small to de- tect signal from FTIR spectroscopy, we found it exhibited a self-similar recursive energy spectrum characteristic of interference from a moir ́e superlattice. Known as Hofstadter’s butterfly, this recursive energy spectrum is one of the first fractals attributable to quantum effects. We analyze the resulting transport data and verify the Hofstadter spectrum in bilayer graphene