Topological Confinement in Bilayer Graphene

We study a new type of one-dimensional chiral states that can be created in bilayer graphene (BLG) by electrostatic lateral confinement. These states appear on the domain walls separating insulating regions experiencing the opposite gating polarity. While the states are similar to conventional solitonic zero modes, their properties are defined by the unusual chiral BLG quasiparticles, from which they derive. The number of zero mode branches is fixed by the topological vacuum charge of the insulating BLG state. We discuss how these chiral states can manifest experimentally and emphasize their relevance for valleytronics.

Figure 1

Side view of a gated bilayer graphene configuration with the voltage kink. The region where the interlayer voltage changes sign (channel) supports bands of chiral zero modes (dashed line). The conventional (nontopological) confinement would correspond to the same polarity of bias on both sides of the channel.

Figure 2

Spatial structure of the confined state obtained numerically from Eqs. (5, 6) for particular value $p_y=-1/2^{3/4}$. The energy of the state is $\epsilon =0.1140$. Shown are the components $u$ and $v$ of the wave function (thin and dashed lines, respectively), the probability density $|\Psi {|}^2=u^2+v^2$ (thick line), and the potential profile, taken here as $\phi (x)=\tanh (x)$. Note the symmetry, $u(x)=v(-x)$.

Figure 3

Energy level structure in the presence of nonuniform interlayer bias $\phi (x)$. The low-energy states in (a) and (c) are localized on the single kink of profile $\phi (x)=\tanh (x/\ell )$ with $\ell =0.5$ and $\ell =4$, respectively. For (a) and (c) we used open boundary conditions, $\Psi (\pm 20)=0$, which caused flat bands at $\epsilon =\pm 1$. The dashed lines are the analytical expressions for the intragap state dispersion, Eq. (14). (b) Same as (a) but with kink and antikink at $x=\pm 10$ and periodic boundary conditions. (d) Nontopological confinement for potential profile $\phi (x)={tanh}^2(x/4)$. Note the absence of chiral zero modes.