Ab initio quantum transport through armchair graphene nanoribbons: Streamlines in the current density

We calculate the local current density in pristine armchair graphene nanoribbons (AGNRs) with varying width, $N_{\mathrm{C}}$, employing a density-functional-theory-based ab initio transport formalism. We observe very pronounced current patterns (“streamlines”) with threefold periodicity in $N_{\mathrm{C}}$. They arise as a consequence of quantum confinement in the transverse flow direction. Neighboring streamlines are separated by stripes of almost vanishing flow. As a consequence, the response of the current to functionalizing adsorbates is very sensitive to their placement: adsorbates located within the current filaments lead to strong backscattering, while adsorbates placed in other regions have almost no impact at all.

Figure 1

(a) Structure of a hydrogen-terminated AGNR11 and (b) the corresponding orientation of the Brillouin zone of the honeycomb lattice with the $\mathbf{K}$ points $\mathbf{K}/{\mathbf{K}}^{\prime }=\frac{2\pi }{3a}\left(\sqrt{3},\pm 1\right)$.

Figure 2

Transmission of (a) AGNR11 to (d) AGNR14. The arrows indicate the current patterns' energies in Fig. 3. The reference energy $E_{\mathrm{F}}$ is the chemical potential of the isolated, charge-neutral species.

Figure 3

Absolute value of the current density (per bias) associated with a fully transparent channel in a plane 0.5 Å above the ribbon plane (exact energy $E=E_{\mathrm{F}}+0.5\mathrm{eV}$, so $T\left(E\right)=1$ for all ribbons; see Fig. 2). The current flows in the horizontal direction, as marked by the arrows. We checked that current patterns are identical for different energies $E$ with $T\left(E\right)=1$. The calculated transmission $T\left(E\right)$ is depicted in Fig. 2. Along the line in AGNR14, the current will be plotted for AGNRs$(3m-1)$ in Fig. 7.

Figure 4

Absolute value of the current density (per bias) in an AGNR11 at $T=1$ perpendicular to the ribbon plane between the white points in Fig. 3 using the color scale of Fig. 3.

Figure 5

Cone with discrete $k_{y,n}$ points near the Dirac point $\mathbf{K}$ exemplary for an AGNR12. The states of this nanoribbon near the Fermi level are arranged on the hyperbolas with $k_{y,4}=12/13K_y$ and $k_{y,5}=15/13K_y$ due to Eq. (4). So $k_{y,4}$ lies uniquely closest to the Dirac point, $\tilde{n}=4$.

Figure 6

Box with transversal density ${\left|\psi \right|}^2\propto {sin}^2\left(k_{y,\tilde{n}}y\right)$ of electronic states in a box with a wavelength according to Eq. (5). Charge carriers near the Fermi level have to jump to states with such an envelope when crossing an AGNR.

Figure 7

$x$ component of the current density (per bias) scaled by the number of streamlines $m$ at $E=E_{\mathrm{F}}+0.5$ eV along the line sketched in the AGNR14 in Fig.  3 and analogously for AGNRs$(3m$$-$1) with $N_{\mathrm{C}}=5$ (dark blue line), 8 (magenta line), 11 (green line), and 17 (cyan line). Red ticks show node positions; see Figs. 3 and 6. Negative currents near nodal lines indicate small current eddies with backflow.

Figure 8

Absolute value of the current density in an AGNR11 with two transmission channels, $T=2$ (again, in a plane $0.5\AA$ above the ribbon, exact energy $E=E_{\mathrm{F}}+1.0$ eV). The observed pattern follows from a superposition of the channel at $T=1$ in Fig. 3 and an additional channel with a transversal wavelength $\lambda \ne 3a$. Similar to AGNRs$(\ne 3m-1)$ at $T=1$, streamlines survive at the edges but start to mix in the bulk.

Figure 9

Transmission of defected AGNR11 and the pristine case (dashed) for different OH positions from Fig. 3. OH is positioned (a) outside and (b) inside a streamline.

Figure 10

LDOS in a plane $0.5\AA$ above the ribbon plane [exact energy $E=E_{\mathrm{F}}+0.5\mathrm{eV}$, so $T\left(E\right)=1$ for all ribbons; see Fig. 2]. We checked that the LDOS patterns are identical for energies $E$ with $T\left(E\right)=1$ except for their amplitude being dependent on the exact energy. Due to the close AGNR13 van Hove singularity (at $\approx$$E_{\mathrm{F}}+0.4\mathrm{eV}$), the LDOS at $E=E_{\mathrm{F}}+0.5$ eV is enhanced in AGNR13 compared to the other AGNRs.

Figure 11

LDOS of an AGNR11 in a plane 2 Å above the ribbon plane at $E=E_{\mathrm{F}}+0.5$ eV.