Moire bands in twisted double-layer graphene (1009.4203v1)

Abstract: A moire pattern is formed when two copies of a periodic pattern are overlaid with a relative twist. We address the electronic structure of a twisted two-layer graphene system, showing that in its continuum Dirac model the moire pattern periodicity leads to moire Bloch bands. The two layers become more strongly coupled and the Dirac velocity crosses zero several times as the twist angle is reduced. For a discrete set of magic angles the velocity vanishes, the lowest moire band flattens, and the Dirac-point density-of-states and the counterflow conductivity are strongly enhanced.

• The paper develops a continuum Dirac model to reveal how twisting induces moiré Bloch bands and causes the Dirac velocity to vanish at magic angles.
• The study employs plane-wave expansions to numerically capture band structure variations, showing renormalization effects near van Hove singularities.
• The observed flat band at approximately 1.05° boosts the density of states and counterflow conductivity, highlighting potential for innovative electronic applications.

Moiré Bands in Twisted Double-Layer Graphene: A Summary

This paper, authored by R. Bistritzer and A.H. MacDonald, presents a rigorous theoretical investigation into the electronic properties of twisted double-layer graphene, focusing on the formation of moiré patterns and their impact on electronic band structures. Utilizing a continuum Dirac model, the paper elaborates on how the twist between graphene layers induces moiré Bloch bands, which bear significant implications for the electronic phenomena observed in these materials.

Key Insights and Findings

The authors explore the electronic structures of twisted bilayer graphene, specifically identifying the influence of twist angles on the coupling between layers. As the twist angle decreases, the layers become more strongly coupled, resulting in the Dirac velocity crossing zero at discrete "magic" angles. This leads to a flattening of the lowest moiré band and a substantial increase in both the Dirac-point density-of-states (DOS) and counterflow conductivity.

1. Twist Angle and Continuum Model: The paper constructs a low-energy continuum model Hamiltonian, combining single-layer Dirac-Hamiltonian terms and a tunneling term that models interlayer coupling. This approach allows for computationally feasible scrutiny of band structures across arbitrary angles.
2. Band Structure Calculations: The authors numerically solve for the moiré bands using plane-wave expansions, demonstrating that for large twist angles, the moiré band structure closely resembles that of isolated graphene, except for velocity renormalization at high energies near van Hove singularities.
3. Magic Angles: One of the striking results is the observation of a vanishing Dirac-point velocity at approximately 1.05°, a behavior attributed to a flattening of the band that enhances the DOS at the Dirac point. This vanishing is not monotonic, and magic angles recur at other minor twist angles, which adds complexity to the electronic properties of the system.
4. Exotic Transport Effects: An intriguing observation is the counterflow conductivity's behavior. Despite the flat band at certain twist angles leading to a large DOS, there is no concomitant reduction in counterflow velocity, which effectively boosts the counterflow conductivity due to increased carrier density without velocity decrease.
5. Unresolved Phenomena: The paper acknowledges certain unresolved aspects, such as explaining the complete flattening of the lowest moiré band at 1.05°, and predicts further interest in exploring systems with more than two graphene layers. Electron-electron interactions, neglected in their present analysis, are anticipated to play a substantial role at magic angles, potentially resulting in novel quantum phenomena like counterflow superfluidity and flat-band magnetism.

Implications and Future Directions

From a practical perspective, understanding moiré band structures has significant implications for developing novel electronic and optoelectronic devices utilizing graphene bilayers. The paper's findings on twist induced variations in electronic properties provide a foundation for engineering material systems with tailor-fit characteristics for specific applications.

Theoretically, this work paves the way for deeper exploration of many-body effects in twisted bilayer graphene, especially under conditions where electron interactions dominate. Subsequent research will likely focus on multidimensional graphene systems and the role of interactions at magic angles, potentially unraveling new quantum states of matter.

In conclusion, the paper delivers substantial contributions to the field of condensed matter physics by elucidating the complex interplay between moiré patterns and electronic properties in twisted bilayer graphene, setting a stage for experimental verification and application-focused innovations.