Spin-orbit coupling and broken spin degeneracy in multilayer graphene

TL;DR

This paper investigates how spin-orbit coupling causes spin splitting in multilayer graphene, especially in trilayer graphene, due to broken inversion symmetry, revealing a Zeeman-like effect with an estimated effective magnetic field of about 2 Tesla.

Contribution

The study identifies a new intrinsic spin-orbit coupling in trilayer graphene that induces spin splitting, extending understanding beyond Kane-Mele type interactions.

Findings

Spin splitting occurs in odd-layer graphene due to broken inversion symmetry.

A second type of intrinsic spin-orbit coupling introduces Zeeman-like splitting.

Estimated effective magnetic field from spin splitting is approximately 2 Tesla.

Abstract

Since the lattices of ABA-stacked graphene multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial-inversion symmetry, their electronic bands must be spin degenerate in the presence of time-inversion symmetry. In intrinsic monolayer and bilayer graphene, when symmetry is not broken by external fields, the only spin-orbit coupling present at low energy near the corner of the Brillouin zone is the Kane-Mele term, that opens a bulk energy gap but does not break the spin degeneracy of the energy bands [C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)]. However, spin splitting is allowed in multilayers with an odd number of layers (greater than or equal to 3) because their lattices do not satisfy spatial inversion symmetry. We show that, in trilayer graphene, in addition to the Kane-Mele term, there is a second type of intrinsic spin-orbit coupling present at low energy near the corner of the Brillouin zone. It introduces a Zeeman-like spin splitting of the energy bands at each valley, with an opposite sign of the effective magnetic field in the two valleys. We estimate the magnitude of the effective field to be ~2T.