Andreev reflection in graphene nanoribbons

We study Andreev reflection in graphene nanoribbon/superconductor hybrid junctions. By using a tight-binding approach and the scattering formalism we show that finite-size effects lead to notable differences with respect to the bulk-graphene case. At subgap voltages, conservation of pseudoparity, a quantum number characterizing the ribbon states, yields either a suppression of Andreev reflection when the ribbon has an even number of sites in the transverse direction or perfect Andreev reflection when the ribbon has an odd number of sites. In the former case the suppression of Andreev reflection induces an insulating behavior even when the junction is biased; electron conduction can however be restored by applying a gate voltage. Finally, we check that these findings remain valid also in the case of nonideal nanoribbons in which the number of transverse sites varies along the transport direction.

Figure 1

(Color online) A hybrid junction between a zigzag graphene nanoribbon and a superconductor.

Figure 2

Coordinates and elementary cells in GNRs in the cases where the number $N_W$ of sites in the transverse direction is (a) even and (b) odd.

Figure 3

(Color online) The band structure of a GNR as a function of the longitudinal wave vector $k_x$. Only the three lowest bands are illustrated. Energies are in units of $\gamma$. The solid (dashed) lines represent the particle (hole) bands. In this figure $U=0$ (zero gate voltage). In this case particle and hole bands are degenerate (for the sake of clarity the hole bands have been slightly shifted to the right). The signs $+$ and $-$ inside the circles indicate the value $\eta$ of pseudoparity for each band. (a) $N_W$ is even. In the range $0<\epsilon <\delta$ no AR is possible since left-moving hole states 2 have pseudoparity opposite to the one of incoming right-moving particle states 1. In contrast, for $\epsilon >\delta$, left-moving hole states 4 and 5 with the same pseudoparity as incoming particle states 3 are available, and AR is finite. (b) $N_W$ is odd. In this case AR is possible in the range $0<\epsilon <\delta$ since left-moving hole states 2 have the same pseudoparity of incoming right-moving particle states 1. For $\epsilon >\delta$ left-moving hole states 4 and 6 with the same pseudoparity as incoming particle states 3 are available and AR is again possible. Note that left-moving hole state 5 in this case has pseudoparity opposite to the one of incoming right-moving particle states 3.

Figure 4

(Color online) Same as in Fig. 3 but for $U\ne 0$ (finite gate voltage). In this case for $\epsilon <U$ incoming particle states and outgoing hole states have the same pseudoparity, and AR is allowed.

Figure 5

(Color online) Transport properties of a GNR/SC interface. The differential conductance $G$ in units of $2e^2/h$ (solid line), the total Andreev-reflection coefficient $R_A$ (dashed line), and the number $N$ of open channels (dotted line) are shown as functions of the applied bias $eV$ (in units of ${\Delta }_0$). Note that the horizontal axis has been broken in two ranges. The GNR has a width of about 50 nm, which corresponds to $\delta \sim 52\text{meV}=52{\Delta }_0$ (the superconducting gap has been fixed at the value ${\Delta }_0=1\text{meV}$ and the hopping energy $\gamma$ to the value 2.8 eV). (a) $N_W=250$. The AR coefficient is nonzero only for $eV>\delta$ and has been multiplied by a factor 500 for clarity. (b) $N_W=251$. The AR coefficient is unity for $eV<{\Delta }_0$, drops abruptly to zero for $eV>{\Delta }_0$, and remains zero up to $eV=\delta$. The AR coefficient is again finite for $eV>\delta$ and has been multiplied by a factor 200 for clarity.

Figure 6

(Color online) Same as in Fig. 5 but for $U=0.5{\Delta }_0$ (in this figure we have not plotted the number of open channels). (a) $N_W$ is even. In this case the AR coefficient is restored to a finite value for $eV<U$, while it remains zero for $U<eV<\delta$. (b) $N_W$ is odd. In this case no big qualitative changes are seen to occur by switching on a finite value of $U$.

Figure 7

(Color online) Schematic representation of a GNR with nonideal edges.

Figure 8

(Color online) The differential conductance $G$ in units of $2e^2/h$ is shown as a function of the applied bias $eV$ (in units of ${\Delta }_0$) for various values of $\overline{M}$. (a) $N_{W,i}=100$. The (black) short-dashed line corresponds to $\overline{M}=4$, whereas the (red) long-dashed line and the (green) solid line correspond to $\overline{M}=6$ and $\overline{M}=8$, respectively. (b) Same as in panel (a) but for $N_{W,i}=101$. For $\overline{M}=8$ the ideal-GNR behavior depicted in Fig. 6 is almost recovered.

Figure 9

(Color online) The threshold length $M^{\star }$ as a function of $N_{W,i}$ (even case only).