Band-Selective Filter in a Zigzag Graphene Nanoribbon

Electric transport of a zigzag graphene nanoribbon through a steplike potential and a barrier potential is investigated by using the recursive Green’s function method. In the case of the steplike potential, we demonstrate numerically that scattering processes obey a selection rule for the band indices when the number of zigzag chains is even; the electrons belonging to the “even” (“odd”) bands are scattered only into the even (odd) bands so that the parity of the wave functions is preserved. In the case of the barrier potential, by tuning the barrier height to be an appropriate value, we show that it can work as the “band-selective filter”, which transmits electrons selectively with respect to the indices of the bands to which the incident electrons belong. Finally, we suggest that this selection rule can be observed in the conductance by applying two barrier potentials.

Figure 1

(a) The band structure of a zigzag ribbon for the case of $N=30$. $2\Delta \simeq 3\tau \pi /N$ is the energy separation between the top of the next highest valence band and the bottom of the next lowest conduction band. (b) A schematic diagram of a zigzag ribbon. The electrostatic potential varies in the shaded region. The $x$ axis ($y$ axis) is taken to be perpendicular (parallel) to graphene lead lines. (c) A schematic diagram of the scattering process from $K_0$ into $K_{-2}^{\prime }$ (see text). The bands are indexed as shown in the diagram. The colored regions shown in the diagram correspond to the ones shown in Figs. 4, 5.

Figure 2

The transmission probabilities for (a) $\nu =K_0$ and (b) $\nu =K_1$ with $N=30$. The insets are the enlargements of the low-probability regions. Although we do not show $T(K_2^{\prime },K_0)$, $T(K_4,K_0)$, $T(K_4^{\prime },K_0)$ and $T(K_3^{\prime },K_1)$ in the diagram, they have a very small yet nonzero value (less than $\sim {10}^{-4}$), respectively. The others ($T(K_1,K_0)$, $T(K_2,K_1)$, etc.) are suppressed to 0. It can be seen that intravalley scatterings are not always dominant compared to intervalley scatterings, e.g., $T(K_4,K_0)<T(K_4^{\prime },K_0)$.

Figure 3

(a) A schematic diagram of a zigzag ribbon for even $N$. Open and filled circles represent two inequivalent sublattices, respectively. One can regard a zigzag ribbon as a quasi-one-dimensional system. In this regime, the columns $\alpha$ and $\beta$ are treated as quasi sublattices. (b),(c) The wave functions can be written as ${\Psi }_{K_n(K_n^{\prime })}(x,y)={\psi }_{K_n(K_n^{\prime })}(x)e^{i{k}_{y}y}$. We show ${\psi }_{K_n(K_n^{\prime })}(x)$ of (b) the incoming channels $K_0$, $K_1$ on the quasi sublattice $\alpha$ in the left lead for $E_I=0.5\tau$, (c) the outgoing channels $K_{-1}^{\prime }$, $K_{-2}$, $K_{-3}$ and $K_{-4}$ on $\alpha$ in the right lead for $\epsilon =-0.5\tau$. The wave functions on the quasi sublattice $\beta$ have the same parity as the one of the wave functions on $\alpha$. Here $N=30$.

Figure 4

$P_e$ and $P_o$ for the clean system (solid line) and the disordered system (dashed line) with $L_0=50a$. The impurities are distributed in the range $y\in [-50a,100a]$ and each dashed line is an average over 10 realizations of random potentials. The integral ratios shown in the left (right) panel represent the values of ${\mathcal{N}}_e/\mathcal{N}$ (${\mathcal{N}}_o/\mathcal{N}$) (see text). The colored regions shown in the diagram correspond to the ones shown in Fig. 1c.

Figure 5

(a) Schematic diagrams of band structures and the barrier potential described as Eq. (5). (b) The conductances for the clean system (solid line) and the disordered system (dashed line) with $L_0=L_2=50a$ and $L_1=100a$. The impurities are distributed in the range $y\in [-50a,250a]$ and each dashed line is an average over 10 realizations of random potentials. In the left (right) panel, $V_0$ is tuned to be $0.426\tau$ ($0.574\tau$) so that $\epsilon$ is in the even (odd) pass region, and thus the odd (even) pass region in the second potential turns to the pseudogap region. In the region ${\epsilon }^{\prime }<0$, only interband scatterings occur, which makes the behavior of the conductances complicated.