Conductance of electrostatic wire junctions in bilayer graphene

The conductance of electrostatic wire junctions in bilayer graphene, classified as trivial-trivial or trivial-topological regarding the confinement character on each junction side, is calculated. The topological side always corresponds to a kink-antikink system, as required for a proper connection with a trivial side. We report a conductance quench of the trivial-topological junction, with a conductance near quantization to $4e^2/h$, which is only half of the maximum value allowed by the Chern number of a kink-antikink system. The analysis allowed us to uncover the existence of a chiral edge mode in the trivial wire under quite general conditions. A double junction, trivial-topological-trivial, displays periodic Fano-like conductance resonances (dips or peaks) induced by the created topological loop.

Figure 1

Sketches representing BLG electrostatic junctions of (a) trivial-trivial and (b) trivial-topological type. The graphene layers are the gray planes, and the top and bottom microelectrodes are the colored planes. Blue and red indicate different signs of the applied potential on the corresponding microelectrode. The white dashed line indicates the junction position. The lateral widths on each junction side are $L_y$ and $L_y^{\prime }$. Bulk and edge electronic modes incident from the left are indicated by thick and thin white arrows, respectively. Notice that in (b), bulk modes are mostly backscattered and only the edge chiral modes discussed in this paper are transmitted from left to right.

Figure 2

Conductance of electrostatic single junctions (green) of trivial-trivial type (a) and trivial-topological type (b). Black and red data show the number of propagating modes (Chern number) on the left and right sides of the junction, labeled as ${\mathcal{N}}_{\mathrm{tri}}, {\mathcal{N}}_{\mathrm{tri}}^{\prime }$ (a) and ${\mathcal{N}}_{\mathrm{tri}}, {\mathcal{N}}_{\mathrm{top}}^{\prime }$ (b), respectively. Parameters: $L_y=600$ nm, $L_y^{\prime }=400$ nm, $|V_a|=20$ meV, and $s=12$ nm.

Figure 3

(a) Energy bands of the trivial confinement wire with $L_y=600$ nm, $|V_a|=20$ meV, and $s=12$ nm. The shaded region corresponds to the existence of real-$q$ modes, $k<k_c$ with $k_c$ being the critical value given in Eq. (11). (b)–(d) Probability densities for the $k$ values of the successive bands indicated with red dots in (a).

Figure 4

(a) Sketch of a semi-infinite interface between unbiased and biased BLG. The arrows indicate the propagation of the edge chiral mode for $K$ and $K^{\prime }$ valleys along the interface shown with a dashed line. (b) Energy dispersion of the edge chiral modes. The shading corresponds to the continuum of states for $k<k_c$ of Eq. (11). Parameters: $|V_a|=20$ meV and $s=12$ nm.

Figure 5

(a) and (b) Sketches of the double-junction setups similar to Fig. 1. The length of the central part is $L_x=1\mu \mathrm{m}$. (c) and (d) Results of the double junctions shown in (a) and (b), respectively. The black line indicates the number of modes in trivial left and right leads, while the red line indicates the number of modes in the center, which can be either trivial [(a) and (c)] or topological [(b) and (d)]. The rest of the parameters are as in Fig. 2.

Figure 6

(a) Energy dispersion of the $K$ and $K^{\prime }$ edge chiral mode for the semi-infinite interface [Fig. 4] and for two different values of the diffusivity parameter $s$. We have assumed $V_a=20$ meV, and the shaded region corresponds to $k<k_c$. (b) Spatial distribution of the probability densities for the states indicated in (a). The solid curves in (a) and (b) are the results of the quasianalytic model for a sharp interface, and they show an excellent agreement with the $s=0.5$ nm smooth potential data.

Figure 7

Same as Fig. 6 for different values of the asymmetry potential $V_a$. We assumed a sharp interface $s=0.5$ nm.