Chiral orbital current and anomalous magnetic moment in gapped graphene

We present a low-energy effective-mass theory to describe a chiral orbital current and an anomalous magnetic moment in graphenes with a band gap and related materials. We explicitly derive a quantum-mechanical current distribution in general Bloch electron systems, which describes a chiral current circulation supporting the magnetic moment. We apply the formulation to gapped graphene monolayer, bilayer, and ABC-stacked multilayers to show that the chiral current is opposite between different valleys, and the corresponding magnetic moment accounts for valley splitting of Landau levels. In a gapped bilayer and ABC multilayer graphenes, in particular, the valley-dependent magnetic moment is responsible for huge paramagnetic susceptibility at low energy, which enables a full valley polarization up to relatively high electron densities. The formulation also applies to the gapped surface states of a three-dimensional topological insulator, where the anomalous current is related to the magnetoelectric response in a spatially modulated potential.

Figure 1

(a) Low-energy dispersion of gapped bilayer graphene given by Eq. (27). (b) Landau level spectrum of Eq. (39) with some small $n$’s, plotted against the magnetic field. Dashed (red) and solid (black) lines represent the valleys $\xi =+$ and $-$, respectively. Numbers assigned to the curves indicate Landau level index $n$. A pair of dotted slopes represents the energy of the band bottom shifted by pseudospin Zeeman energy, i.e., $-{\varepsilon }_0\pm \hslash {\omega }_0/2$. At $\Delta =0.1\mathit{eV}$, the characteristic energy scale is ${\varepsilon }_0=13\mathrm{meV}$, and the magnetic field for $\hslash {\omega }_0/{\varepsilon }_0=1$ is 7.6 T.

Figure 2

(a) Low-energy dispersion of gapped three- and four-layer ABC graphene given by Eq. (44). (b) and (c) Corresponding Landau level spectrum of Eq. (52) with some small $n$’s, plotted against the magnetic field. A pair of dotted slopes represents $-{\varepsilon }_0\pm \hslash (N-1){\omega }_0/2$.

Figure 3

Susceptibility (solid line) and density of states (dashed line) near the band bottom of asymmetric bilayer graphene, plotted against the Fermi energy.

Figure 4

Single-valley current distribution contributed by conduction-band electrons of gapped monolayer (solid curve) and bilayer graphenes (dashed curve), terminated at $x=0$.