Helicoidal graphene nanoribbons: Chiraltronics

We present a calculation of the effective geometry-induced quantum potential for the carriers in graphene shaped as a helicoidal nanoribbon. In this geometry the twist of the nanoribbon plays the role of an effective transverse electric field in graphene and this is reminiscent of the Hall effect. However, this effective electric field has a different sign for the two isospin states and translates into a mechanism to separate the two chiral species on the opposing rims of the nanoribbon. Isospin transitions are expected with the emission or absorption of microwave radiation which could be adjusted to be in the THz region.

Figure 1

Two helicoidal nanoribbons with different chiralities: (a) $\omega >0$ and (b) $\omega <0$. The vertical axis is along $x$ and the transverse direction $\xi$ is across the nanoribbons. Here ${\xi }_0$ is the inner radius and $D$ is the outer radius. The two graphene isospin states (color coded as red and blue) collect on opposing rims (separated in space). The respective rims are exchanged when the chirality of the helicoid is reversed. The same exchange takes place when the direction of propagation along the helicoid changes, that is, $m\to -m.$

Figure 2

The potential acting on each of the isospin states as a function of the width of the helicoid $\xi$. Here $\omega >0.$ Note that the potentials have a maximum and then fall off $\propto 1/{\xi }^2.$ The extremum for $\left|m\right|=1$ state is reached for ${\xi }_{\mathrm{extr}}=1/\left(\omega \sqrt{8}\right).$ For $\xi \gg {\xi }_{\mathrm{extr}}$ the isospin separation scales as $\mathrm{\Delta }U(\xi \gg {\xi }_{\mathrm{extr}})\approx \frac{2\left|m\right|}{{\xi }^2}.$

Figure 3

Provided the nanoribbon is small enough, so that $\xi <{\xi }_{\mathrm{extr}},$ the potential acting on each of the isospin states as a function of the width of the helicoid scales linearly with $\xi .$ Note that the difference between the potentials acting on the two isospin states is $\mathrm{\Delta }U(\xi <{\xi }_{\mathrm{extr}})\approx 2\left|m\right|{\left|\omega \right|}^3\xi .$ The frequency of the expected transition is in the THz region (for micron-sized ribbons). See text for further details.