Effect of zigzag and armchair edges on the electronic transport in single-layer and bilayer graphene nanoribbons with defects

We study electronic transport in monolayer and bilayer graphene with single and many short-range defects focusing on the role of edge termination (zigzag versus armchair). Within the tight-binding approximation, we derive analytical expressions for the transmission amplitude in monolayer graphene nanoribbons with a single short-range defect. The analytical calculations are complemented by exact numerical transport calculations for monolayer and bilayer graphene nanoribbons with a single and many short-range defects and edge disorder. We find that for the case of the zigzag edge termination, both monolayer and bilayer nanoribbons in a single- and few-mode regime remain practically insensitive to defects situated close to the edges. In contrast, the transmission of both armchair monolayer and bilayer nanoribbons is strongly affected by even a small edge defect concentration. This behavior is related to the effective boundary condition at the edges, which, respectively, does not and does couple valleys for zigzag and armchair nanoribbons. In the many-mode regime and for sufficiently high defect concentration, the difference of the transmission between armchair and zigzag nanoribbons diminishes. We also study resonant features (Fano resonances) in monolayer and bilayer nanoribbons in a single-mode regime with a short-range defect. We discuss in detail how an interplay between the defect's position at different sublattices in the ribbons, the defect's distance to the edge, and the structure of the extended states in ribbons with different edge termination influence the width and the energy of Fano resonances.

Figure 1

(a) Honeycomb lattice structure of graphene composed of two interpenetrating triangular lattices; ${\mathbf{a}}_1=a(3/2,\sqrt{3}/2)$ and ${\mathbf{a}}_2=a(3/2,-\sqrt{3}/2)$ are the lattice unit vectors, and $a\approx 1.42\AA$ is the carbon-carbon distance.[36] $A$ (red) and $B$ (blue) mark two sublattices of the graphene lattice. (b) Asymmetric graphene sheet. (c) Labeling of carbon atoms in the rectangular unit cell used in analytical calculations. (d) Structure of bilayer graphene indicating sublattices $A_1$,$B_1$ (upper layer) and $A_2$,$B_2$ (lower layer) and hopping integrals ${\gamma }_1$ between sites in the sublattices $A_1$ and $A_2$, and ${\gamma }_3$ between sites in the sublattices $B_1$ and $B_2$.

Figure 2

Transmission of the monolayer and bilayer nanoribbons with a single defect at the edge as a function of energy. Insets indicate defect positions. Red and blue curves correspond to a defect in sublattices $A$ and $B$, respectively. Green curves indicate the transmission probability for ideal nanoribbons (without defects). Transmission curves are slightly shifted for clarity in the vertical direction.

Figure 3

Transmission of the monolayer and bilayer nanoribbons with a single defect as a function of the defect's position across the nanoribbons in a single- and many-mode regime [left and right panels in (a) and (b)]. Red and blue curves correspond to a defect in sublattices $A$ and $B$, respectively. For bilayer nanoribbons, the defect is in the upper layer. Insets indicate the numbering of sites.

Figure 4

LDOS for monolayer and bilayer zigzag and armchair nanoribbons in the energy interval corresponding to the first propagating mode. (a) Ideal zGNR terminated by $B$ atoms at the lower edge. (b) A defect in sublattice $B$ at the edge of zGNR at the energy corresponding to the resonant dip. (c) A defect in sublattice $A$ close to the edge of zGNR. [The values of LDOS in (b) are $\sim 1000$ times higher in comparison to (a) and (c).] (d) Ideal aGNR; (e) Ideal aBGN (LDOS is indicated for the upper layer). Empty circles in (d) and (e) correspond to zero LDOS.

Figure 5

Resonant dips in the transmission of a (a) zGNR, (b) zBGN, (c) aGNR, and (d) BGN with a single defect situated at different distances from the ribbon's edge as indicated in the figures. Red and blue curves correspond to the defect on the $A$ and $B$ sublattices. The number of sites across the nanoribbons is $M=40$ for zGNR and zBGN (corresponding to the width $w=4.12$ nm), and $M=20$ for aGNR and aBGN (corresponding to the width $w=2.34$ nm).

Figure 6

LDOS of a zigzag nanoribbon with a single defect at the sublattice $A$ at the resonant energy corresponding to the transmission dip ($E=0.216544$). Defect position is (a) $n=1$, (b) $n=5$ (site numbers at the $x$ and $y$ axis and the numbering of the defect position are the same as in the previous figures). The ribbon's width is $M=40$ sites.

Figure 7

Transmission of monolayer and bilayer nanoribbons with disordered edges. (a) zGNR, (b) aGNR, (c) zBGN, and (d) aBGN. Top insets illustrate two models of disordered edges used in the present study. Simulations are averaged over 50 realizations.