Valley Symmetry Breaking in Bilayer Graphene: A Test of the Minimal Model

Physical properties reflecting the valley asymmetry of Landau levels in a biased bilayer graphene under a magnetic field are discussed. Within the 4-band continuum model with a Hartree-corrected self-consistent gap and finite damping factor we predict the appearance of anomalous steps in quantized Hall conductivity due to the degeneracy lifting of Landau levels. Moreover, the valley symmetry breaking effect appears as a nonsemiclassical de Haas–van Alphen effect where the reduction of the oscillation period to half cannot be accounted for through quasiclassical quantization of the orbits in reciprocal space.

Figure 1

Landau-level asymmetry at $K$ and $K^{\prime }$ points in bilayer graphene with energy difference $\Delta$.

Figure 2

(a) $\Delta$ vs $n$ for different values of magnetic field. The inset shows the case of a fixed top gate. (b) Magnetization vs $n$ for different values of magnetic field. The vertical lines signal the semiclassical period.

Figure 3

Hall conductivity vs electron density at different magnetic fields: $B=10\mathrm{T}$ (a), 20 T (b), 30 T (c).