Quantum Valley Hall Effect, Orbital Magnetism, and Anomalous Hall Effect in Twisted Multilayer Graphene Systems

Abstract

We study the electronic structures and topological properties of (M+N)-layer twisted graphene systems. We consider the generic situation thatN-layer graphene is placed on top of the otherM-layer graphene and is twisted with respect to each other by an angleθ. In such twisted multilayer graphene systems, we find that there exist two low-energy flat bands for each valley emerging from the interface between theMlayers and theNlayers. These two low-energy bands in the twisted multilayer graphene system possess valley Chern numbers that are dependent on both the number of layers and the stacking chiralities. In particular, when the stacking chiralities of theMlayers andNlayers are opposite, the total Chern number of the two low-energy bands for each valley equals±(M+N−2)(per spin). If the stacking chiralities of theMlayers and theNlayers are the same, then the total Chern number of the two low-energy bands for each valley is±(M−N)(per spin). The valley Chern numbers of the low-energy bands are associated with large, valley-contrasting orbital magnetizations, suggesting the possible existence of orbital ferromagnetism and anomalous Hall effect once the valley degeneracy is lifted either externally by a weak magnetic field or internally by Coulomb interaction through spontaneous symmetry breaking. Such an orbital ferromagnetic state is characterized by chiral current loops circulating around theAAregion of the moiré pattern, which can be experimentally detected.

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Plain Language Summary

Twisted bilayer graphene—two sheets of carbon stacked slightly askew to one another—has drawn significant attention recently because of observations of exotic electrical properties such as superconductivity. Recent theoretical studies have suggested that these phenomena are intimately related to the topological properties of the “flat bands” at the “magic angle”—the amount of twist in the graphene layers around which the electrons’ kinetic energy is maximally quenched. Here, we use analytical arguments and numerical calculations to show that topological flat bands also exist in twisted multilayer graphene, where multiple graphene layers are stacked and rotated.

To characterize the flat bands in multilayer graphene, we calculate their total topological index, a number typically associated with exotic electrical properties such as quantized Hall conductivity. Interestingly, we find that the topological index for the flat bands universally depends on the number and stacking pattern of the twisted layers through a very simple formula, which can be mathematically proved under certain approximations. With the topological structure of these flat bands, a new type of ferromagnetism, known as orbital ferromagnetism, can be generally established in these systems, leading to experimentally measurable current patterns and local magnetic-field distributions.

The topological flat bands and the orbital magnetism make the twisted multilayer graphene systems a unique platform to study strongly correlated physics with nontrivial band topology. The predicted orbital ferromagnetic state is an entirely new state of matter, which may lead to unique transport and optical properties.