To Understand Hofstadter’s Butterflyin Moiré Superlattices

A Fractal Quantum Hall Effect

The Hall Effect

Applications include:• Measuring carrier concentration• Measuring mobility• Magnetic Field Meter• Hall-Effect Multiplier• EtcQuantum Hall effect:• Two dimensional electron system• Produced in a metal-oxide-semiconductor-

field-effect-transistor• Requires high magnetic field and low

temperatures• Quantization of the Hall Effect

Moiré Superlattices

• A moiré pattern is the combination of two or more patterns viewed at the same time.• A superlattice is a periodic structure of layers of two or more

materials. (hBN and Graphene)

Bloch Bands

• Bloch-wave description applies to any wave-like phenomenon in a periodic medium. • Lead to the surprising result that electrons in a conductor scatter only

from imperfections and not from the periodic ions.• Bloch-waves of the moiré superlattice depends on the angular

rotation between the layers of hexagonal Boron Nitride and Bilayer Graphene.

Landau Energy Levels

• Each set of wave functions with the same value of n (Quantum energy number) is called a Landau level. • Effects of Landau levels are only observed when the mean thermal

energy is smaller than the energy level separation • meaning low temperatures and strong magnetic fields.• Each Landau level is degenerate due to the second quantum number

*Degeneracy is how many different states your wave function can take on at a given energy level.

Hofstadter’s Butterfly

• Subject electrons to two forces simultaneously: Magnetic field and periodic Electric field• Pattern of energy levels created

by the two forces is said to be “fractal”• Dark areas of the graph are

where the electrons can be.• White areas are electric band

gaps.

Results

Conclusions• We have to get to work applying these

well executed experiments showing the possibility of BLG as a transistor.

• Nanoelectronics begins here.