Controlling edge states of zigzag carbon nanotubes by the Aharonov-Bohm flux

It has been known theoretically that localized states exist around the zigzag edges of a graphite ribbon and of a carbon nanotube, whose energy eigenvalues are located between conduction and valence bands. We find that in metallic, zigzag single-walled, carbon nanotubes two of the localized states become critical, and that their localization length is sensitive to the mean curvature of a tube, and it can be controlled by the Aharonov-Bohm flux. The curvature-induced mini gap closes by the relatively weak magnetic field. A conductance measurement in the presence of the Aharonov-Bohm flux can give information about the curvature effect and the critical states.

Figure 1

Lattice structure of a zigzag carbon nanotube. The filled (open) circle indicates the $A$ $\left(B\right)$ sublattice. Both the left and right edges are Fujita’s edges.

Figure 2

(a) Region for localized states ${\kappa }^2>4$ shown as the shaded area in the $(g,ϵ)$ plane. Whether localized states are allowed depends on the boundary condition. If the figure is rewritten in terms of $k$ and $E$, the empty region reproduces the well-known band structure for the graphene sheet. (b) When we ignore the curvature effect, there are states with $g=-1$ in the absence of the AB flux. The curvature effect displaces the states onto the line $g=-[1+(\delta V_2-\delta V_1)∕V_{\pi }]$. The AB flux changes the value of $g$, and the eigenstates will make trajectories like the dashed curves in the figure. The two extended states (filled circles) become localized as they run into the shadow region. The eigenstates, except for these two states, remain extended (open circles).

Figure 3

Localization length of the critical state for (a) $n=30$ and (b) $n=51$ zigzag nanotubes. The horizontal axis is a magnetic field, $B\mathrm{T}$, (a) $7.5⩽B⩽30$ and (b) $1.6⩽B⩽30$.

Figure 4

Theoretical result of conductance with a metallic zigzag nanotube. We plot conductance as a function of gate voltage, $V_g$ for several values of the AB flux.