Intrinsic Zeeman Effect in Graphene

TL;DR

This paper investigates the intrinsic Zeeman effect in graphene, revealing how Coulomb interactions lift degeneracies in Landau levels, leading to quantum Hall ferromagnetism and spin-dependent Coulomb potentials across monolayer, bilayer, and trilayer graphene.

Contribution

It provides a detailed analysis of Landau level degeneracy removal in graphene due to Coulomb interactions, highlighting the emergence of quantum Hall ferromagnetism and spin-dependent effects.

Findings

Zero-energy level develops an excitonic gap at ν=0.

Quantum Hall ferromagnetism appears at specific filling factors.

Spin-dependent Coulomb interactions cause valley polarization at certain fillings.

Abstract

The intrinsic Zeeman energy is precisely one half of the cyclotron energy for electrons in graphene. As a result a Landau-level mixing occurs to create the energy spectrum comprised of the $4j$-fold degenerated zero-energy level and 4-fold degenerated nonzero-energy levels in the $j$-layer graphene, where $j=1,2,3$ for monolayer, bilayer and trilayer, respectively. The degeneracy manifests itself in the quantum Hall (QH) effect. We study how the degeneracy is removed by the Coulomb interactions. With respect to the zero-energy level, an excitonic gap opens by making a BCS-type condensation of electron-hole pairs at the filling factor $\nu =0$. It gives birth to the Ising QH ferromagnet at $\nu =\pm 1$ for monolayer, $\nu =\pm 1,\pm 3$ for bilayer, and $\nu =\pm 1,\pm 3,\pm 5$ for trilayer graphene from the zero-energy degeneracy. With respect to the nonzero-energy level, a remarkable consequence is derived that the effective Coulomb potential depends on spins, since a single energy level contains up-spin and down-spin electrons belonging to different Landau levels. The spin-dependent Coulomb interaction leads to the valley polarization at $\nu =\pm 4, \pm 8, \pm 12, ...$ for monolayer, $\nu =\pm 2, \pm 6, \pm 10, >...$ for bilayer, and $\nu =\pm 2,\pm 4, \pm 8, \pm 12, ...$ for trilayer graphene.