Topological confinement in trilayer graphene

We calculate the spectrum of states that are localized at the interface between two regions of opposite bias in trilayer graphene (TLG). These potential profiles, also known as potential kinks, have been predicted to support two different branches of localized states for the case of bilayer graphene, and show similarities to the surface states of topological insulators. On the other hand, we found that ABC stacked TLG exhibits three different unidimensional branches of states in each valley that are confined to the kink interface. They have the property $E\left(k_y\right)=-E(-k_y)$ when belonging to the same valley and $E_K\left(k_y\right)=-E_{K^{\prime }}(-k_y)$. A kink-antikink potential profile opens a gap in the spectrum of these one-dimensional states.

Figure 1

(a) Diagrammatic scheme of couplings in graphene trilayers for ABC stacking. (b) Example of potential profile applied to the bottom and upper layers that support chiral bound states. (c) TLG spectrum with no bias (black) and with bias (red), where $U_2=0$, $U_1=-U_3=100$ meV.

Figure 2

(a) The blue curves are the electronic states for the $K$ valley that are localized at sharp kink for $U_0=100$ meV, while the black curves delimit the cross-hatched free particle regions. (b) The group velocity for the energy branches which are indicated by the numbers $\left(1\right)$, $\left(2\right),$ and $\left(3\right)$ in panel (a).

Figure 3

Probability density of the different spinor components for the zero energy states marked in Fig. 2 by symbols.

Figure 4

Energy spectrum around the $K$ valley for a kink-antikink potential profile with (a) $w=50$ nm and (b) $w=10$ nm separation between the kink and the antikink.