Graphene antidot lattice waveguides

We introduce graphene antidot lattice waveguides: nanostructured graphene where a region of pristine graphene is sandwiched between regions of graphene antidot lattices. The band gaps in the surrounding antidot lattices enable localized states to emerge in the central waveguide region. We model the waveguides via a position-dependent mass term in the Dirac approximation of graphene and arrive at analytical results for the dispersion relation and spinor eigenstates of the localized waveguide modes. To include atomistic details we also use a tight-binding model, which is in excellent agreement with the analytical results. The waveguides resemble graphene nanoribbons, but without the particular properties of ribbons that emerge due to the details of the edge. We show that electrons can be guided through kinks without additional resistance and that transport through the waveguides is robust against structural disorder.

Figure 1

Upper panel: conceptual illustration of a graphene antidot lattice (GAL) waveguide. A central region of pristine graphene is surrounded by GAL regions, the band gaps of which confine states to the waveguide region. Translational symmetry is assumed in the $y$ direction, which is along the longitudinal direction of the waveguide. Lower panel: geometry of a ${\{7,3\}}_2^{\mathrm{zz}}$ GAL waveguide. Black dots show the location of carbon atoms. Bloch boundary conditions are imposed on all boundaries. The dashed lines illustrate the enlarged GAL unit cell, the width of which we denote by $w$. The lowest-energy waveguide mode at the $\Gamma$ point, calculated via the tight-binding model, is illustrated with circles, the size of which shows the absolute value $\left|\psi \right(x,y\left)\right|$ of the (real-valued) eigenstate, while the color indicates the sign.

Figure 2

Left: band structure of the ${\{7,3\}}_5^{\mathrm{zz}}$ GAL waveguide. The band structures are shown for the tight-binding model as well as the analytical infinite mass-limit results $E_{\text{ns}}^{\infty }$ and numerical solution of the transcendental equation of the Dirac approximation $E_{\text{ns}}$. The shaded gray area shows the projected bands of the surrounding GAL regions. For comparison, the bulk graphene band structure $E_b$ is also shown. Note that $\Lambda$ denotes the lattice constant of the waveguide. Right: corresponding density of states for the TB model. Note the Van Hove singularities characteristic of one-dimensional structures.

Figure 3

(a) Energy of the lowest localized waveguide state at the $\Gamma$ point, for the ${\{7,3\}}_N$ family of GAL waveguides. The energy is shown as a function of the width of the waveguide. Results are shown for TB models (points) of waveguides oriented along the zigzag (ZZ) and armchair (AC) directions, respectively, as well as the analytical results obtained via the Dirac model in the infinite mass limit and including the first-order correction (black lines). See the legend in panel (b). Inset: $1/W$ dependence of the energy for wide waveguides. For comparison, the thin colored lines show the energies for ZZ and AC GNRs, if certain edge dependencies are ignored (see text). (b) Corresponding effective masses in units of the free-electron mass $m_e$. In both panels, note the close resemblance of the results obtained for AC and ZZ oriented waveguides.

Figure 4

Wave functions of the localized waveguide modes corresponding to the (a) lowest and the (b) second-lowest (positive) energy at the $\Gamma$ point of a (left) ${\{7,3\}}_5^{(}\mathrm{zz})$ and a (right) ${\{7,3\}}_3^{(}\mathrm{ac})$ GAL waveguide. The lower panels in each case show the geometry, with carbon atoms indicated with black dots. Note that the actual computational cell includes additional GAL unit cells on each side of the central region. Superimposed on top of the geometry is the wave function, with the size of the circles indicating the absolute square of the $\pi$-orbital coefficient, while the color indicates the sublattice. The upper panels show the integrated probability density, $\rho \left(x\right)\equiv {\int \left|\psi (x,y)\right|}^{2}dy$. Red and blue circles indicate the densities on each sublattice. The black line shows the total density, when including a small broadening term. The dashed line shows the corresponding density at nonzero wave vectors, $k\Lambda =\pi /2$. Note the rapid oscillations of the integrated density of the AC waveguide.

Figure 5

Insets: schematic transport setups for (a) a GAL waveguide and (b) a GNR, each connecting two semi-infinite graphene leads. Conductances for pristine (solid lines) and disordered systems (dashed lines) are shown for (a) GAL waveguides and (b) GNRs, in units of the conductance quantum $G_0$. The straight black dashed lines show the graphene conductance. In the disordered systems, edge atoms have been randomly removed with a 5$\%$ probability. The lengths of the waveguides and GNRs are $L=89$nm. The shaded area in panel (a) indicates the energy range of the confined waveguide mode.

Figure 6

Bond current through a “kinked” ${\{5,2\}}_1^{(}\mathrm{zz})$ GAL waveguide. The current is calculated at energy $E=0.25$ eV with a transmission of $\mathcal{T}=1.9$. The current is highly confined to the waveguide region and no additional reflections are observed due to the kinks.

Figure 7

Conductances of the “kinked” waveguide shown in Fig. 6 (straight line) and of a straight waveguide of similar length (dashed line). The shaded area indicates the energy range of the confined waveguide mode of the straight waveguide. Note the nearly identical conductances of the two systems.