Electrons in crystals experience an electric potential with the periodicity in the order of lattice constant, described by Bloch electrons. The band structure of non-interacting two-dimensional electrons is further modified when exposing the lattice in a magnetic field. Back to 1976, Hofstadter firstly calculated the band structure of Bloch electrons in magnetic fields, giving a fractal spectrum, later called Hofstadter butterfly. Experimentally, physicists have struggled to observe this butterfly because a very large magnetic field (in the order of 10⁴ T) is needed due to the small lattice periodicity in conventional crystal lattices.

When two layers of atomically thin materials with similar lattices are stacked together, a periodic long-rang pattern called Moiré superlattice is generated. Such superlattice has been observed in several 2D heterostructures, such as graphene/h-BN and twisted graphene. The large Moiré wavelength (10-100 nm) reduces the magnetic field needed to observe Hofstadter butterfly to about 10 T, which can be easily achieved in normal laboratory. The title image shows the beautiful Hofstadter butterfly observed in magic-angle twisted double bilayer graphene.

Reference:

[1]. L. A. Ponomarenko, et. al., “Cloning of Dirac fermions in graphene superlattices”, Nature, 497, 594-597, (2013).

[2]. C.R. Dean, et. al., “Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices”, Nature, 497, 598-602, (2013).

[3]. B. Hunt, et. al., “Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure”, Science, 340(6139), 1427-1430, (2013).