Edge states on graphene ribbons in magnetic field: Interplay between Dirac and ferromagnetic-like gaps

By combining analytic and numerical methods, edge states on a finite width graphene ribbon in a magnetic field are studied in the framework of low-energy effective theory that takes into account the possibility of quantum Hall ferromagnetism gaps and dynamically generated Dirac-type masses. The analysis is done for graphene ribbons with both zigzag and armchair edges. The characteristic features of the spectrum of the edge states in both these cases are described. In particular, the conditions for the existence of the gapless edge states are established. Implications of these results for the interpretation of recent experiments are discussed.

Figure 1

(Color online) The lattice structure of a finite width graphene ribbon with zigzag (a) and armchair (b) edges.

Figure 2

(Color online) Numerical results for the dimensionless parameters ${\lambda }_{+}$ (solid lines) and ${\lambda }_{-}$ (dashed lines) in the case of ribbons with zigzag edges. The ribbon widths are $W=5l$ (a) and $W=10l$ (b).

Figure 3

(Color online) Numerical results for the low-energy spectra for ribbons with zigzag edges. The ribbon widths are $W=5l$ (a ) and $W=10l$ (b). The ferromagnetic gaps and dynamical masses are as follows: ${\mu }_{\pm }=\mp 0.08{ϵ}_0$, ${\tilde{\mu }}_{\pm }=0.01{ϵ}_0$, ${\Delta }_{\pm }=\pm 0.02{ϵ}_0$, and ${\tilde{\Delta }}_{\pm }=0$. The ferromagnetic gap dominates over the mass gap, insuring the presence of gapless edge states (marked by dots). The electron spins of the lowest-energy sublevels are marked by arrows. The spectra around the $K_{+}\left(K_{-}\right)$ point are shown by solid (dashed) lines.

Figure 4

(Color online) Same as Fig. 3 but with the ferromagnetic gaps and dynamical masses being as follows: ${\mu }_{\pm }=\mp 0.02{ϵ}_0$, ${\tilde{\mu }}_{\pm }=0.01{ϵ}_0$, ${\Delta }_{\pm }=\pm 0.08{ϵ}_0$, and ${\tilde{\Delta }}_{\pm }=0$. The mass gap dominates over the ferromagnetic gap, insuring the absence of gapless edge states.

Figure 5

(Color online) Numerical results for the low-energy spectra for a ribbon with armchair edges of width $W=10l$ in the case of nonzero spin splitting and nonzero singlet masses. Nonzero dynamical parameters are as follows: ${\mu }_{\pm }=\mp 0.02{ϵ}_0$, ${\Delta }_{\pm }=\pm 0.08{ϵ}_0$ (a) and ${\mu }_{\pm }=\mp 0.08{ϵ}_0$, ${\Delta }_{\pm }=\pm 0.02{ϵ}_0$ (b). Gapless edge states (marked by dots) are present in both cases. The electron spins of the lowest-energy sublevels are marked by arrows.

Figure 6

(Color online) Numerical solutions of Eq. (49) for the dimensionless parameter $\lambda =\left[{\left(E+\mu \right)}^2-{\tilde{\Delta }}^2\right]/{ϵ}_0^2$ in the case of armchair edges when $\tilde{\mu }=\Delta =0$. In the higher Landau levels, $n\ge 1$, the solutions to the equations $f_{-}\left(k,\lambda \right)=0$ and $f_{+}\left(k,\lambda \right)=0$ are shown by the solid and dashed lines, respectively.

Figure 7

(Color online) Same as Fig. 5 but for the case of nonzero triplet masses. Nonzero dynamical parameters are as follows: ${\mu }_{\pm }=\mp 0.02{ϵ}_0$, ${\tilde{\Delta }}_{\pm }=0.08{ϵ}_0$ (a) and ${\mu }_{\pm }=\mp 0.08{ϵ}_0$, ${\tilde{\Delta }}_{\pm }=0.02{ϵ}_0$ (b). The existence of gapless modes (marked by dots) depends on the relative magnitude of $\left|{\mu }_{\pm }\right|$ and $\left|{\tilde{\Delta }}_{\pm }\right|$. The electron spins of the lowest-energy sublevels are marked by arrows.