Electrons moving in a
periodic electric potential form Bloch energy bands where the electron masses
are effectively changed. In a strong magnetic field, the cyclotron orbits of
free electrons are quantized and Landau levels form with a massive degeneracy
within. In 1976, Hofstadter showed that for 2-dimensional electronic system,
the intriguing interplay between these two quantization effects can lead into a
self-similar fractal set of energy spectrum known as “Hofstadter’s Butterfly.”
Experimental efforts to demonstrate this fascinating electron energy spectrum
have continued ever since. Recent advent of graphene, where its Bloch electrons
can be described by Dirac fermions, provides a new opportunity to investigate
this half century old problem experimentally. In this presentation, I will
discuss the experimental realization Hofstadter’s Butterfly via substrate
engineered graphene under extremely high magnetic fields controlling two
competing length scales governing Dirac-Bloch states and Landau orbits,
respectively.