Lattice-Induced Double-Valley Degeneracy Lifting in Graphene by a Magnetic Field

We show that the recently discovered double-valley splitting of the Landau levels in the quantum Hall effect in graphene can be explained as the perturbative orbital interaction of intravalley and intervalley microscopic orbital currents with a magnetic field. This effect is facilitated by the translationally noninvariant terms that correspond to graphene’s crystallographic honeycomb symmetry but do not exist in the relativistic theory of massless Dirac fermions in quantum electrodynamics. We discuss recent data in view of these findings.

Figure 1

$K_1$-point Bloch functions ${\Psi }_{K_1}^A$ and ${\Psi }_{K_1}^B$. The $K_2$-point Bloch functions ${\Psi }_{K_2}^A$ and ${\Psi }_{K_2}^B$ are obtained from them by applying complex conjugation.

Figure 2

(a) Two-valley degenerate Dirac-like spectrum $E(k)$ of charge carriers in zero magnetic field, $H=0$. (b) Landau Level (LL) quantization of Dirac Fermions. (c) Orbital valley splitting of LLs. (d) Additional Zeeman spin splitting of LLs (only the $n=0$ level is shown.)

Figure 3

Schematic of circular currents corresponding to Bloch functions ${\Psi }_{\pm }={\Psi }_{K_1}^A\pm {\Psi }_{K_2}^B$ of the split zero Landau level.