Edge states, mass and spin gaps, and quantum Hall effect in graphene

Motivated by recent experiments and a theoretical analysis of the gap equation for the propagator of Dirac quasiparticles, we assume that the physics underlying the recently observed removal of sublattice and spin degeneracies in graphene in a strong magnetic field is connected with the generation of both Dirac masses and spin gaps. The consequences of such a scenario for the existence of the gapless edge states with zigzag and armchair boundary conditions are discussed. In the case of graphene on a half-plane with a zigzag edge, there are gapless edge states in the spectrum only when the spin gap dominates over the mass gap. In the case of an armchair edge, however, the existence of the gapless edge states depends on the specific type of mass gaps.

Figure 1

(Color online) Illustration of the lowest Landau level splitting needed to explain $\nu =0$ and $\nu =1$ plateaus in QHE in graphene.

Figure 2

(Color online) Graphene lattice with zigzag and armchair edges.

Figure 3

(Color online) The numerical solutions of Eq. (34) for the dimensionless parameter ${\lambda }_{+}$ (solid line) and Eq. (36) for the dimensionless parameter ${\lambda }_{-}$ (dashed line) in the case of a zigzag boundary. The solid line at ${\lambda }_{+}=0$ corresponds only to the $E=-{\mu }^{(+)}+{\Delta }^{(-)}$ solution.

Figure 4

(Color online) Numerical results for the energy spectra of the first few Landau levels near a zigzag edge of graphene in the case of nonzero spin splitting and nonzero singlet masses. The values of parameters are ${\mu }_{\pm }=\mp 0.02{ϵ}_0$ and ${\Delta }_{\pm }=\pm 0.08{ϵ}_0$ in the top panel and ${\mu }_{\pm }=\mp 0.08{ϵ}_0$ and ${\Delta }_{\pm }=\pm 0.02{ϵ}_0$ in the bottom panel. (The subscript indices in ${\mu }_{\pm }$ and ${\Delta }_{\pm }$ denote the spin orientations.) In the first case, $\left|{\mu }^{(-)}\right|<\left|{\Delta }^{(+)}\right|$ and there are no gapless modes. In the second case, $\left|{\mu }^{(-)}\right|>\left|{\Delta }^{(+)}\right|$ and the gapless modes are present. The spin-up and spin-down states are denoted by red $\left(s=+\right)$ and blue $\left(s=-\right)$ lines. In the lowest energy sublevels, the spins are also marked by the arrows. The spectra around $K_{+}$ $\left(K_{-}\right)$ point are shown by the solid (dashed) lines.

Figure 5

(Color online) Numerical results for the energy spectra of the first few Landau levels near an armchair edge of graphene in the case of nonzero spin splitting and nonzero singlet masses. The values of parameters are ${\mu }_{\pm }=\mp 0.02{ϵ}_0$ and ${\Delta }_{\pm }=\pm 0.08{ϵ}_0$ in the top panel and ${\mu }_{\pm }=\mp 0.08{ϵ}_0$ and ${\Delta }_{\pm }=\pm 0.02{ϵ}_0$ in the bottom panel. (The subscript indices in ${\mu }_{\pm }$ and ${\Delta }_{\pm }$ denote the spin orientations.) In both cases, there are gapless modes in the spectrum. The spin-up and spin-down states are denoted by red $\left(s=+\right)$ and blue $\left(s=-\right)$ lines. In the lowest energy sublevels, the spins are also marked by arrows.

Figure 6

(Color online) Same as in Fig. 5, but for the case of nonzero triplet masses. The values of parameters are ${\mu }_{\pm }=\mp 0.02{ϵ}_0$ and ${\tilde{\Delta }}_{\pm }=0.08{ϵ}_0$ in the top panel and ${\mu }_{\pm }=\mp 0.08{ϵ}_0$ and ${\tilde{\Delta }}_{\pm }=0.02{ϵ}_0$ in the bottom panel. The existence of gapless modes depends on the relative magnitude of $\left|{\mu }_{\pm }\right|$ and $\left|{\tilde{\Delta }}_{\pm }\right|$.

Figure 7

(Color online) Numerical solutions of Eq. (51) for the dimensionless parameter $\lambda$ in the case of an armchair boundary. This is valid for a general choice of $\tilde{\Delta }$ and $\mu$ but only if $\tilde{\mu }$ and $\Delta$ vanish.