Crossed graphene nanoribbons as beam splitters and mirrors for electron quantum optics

We analyze theoretically four-terminal electronic devices composed of two crossed graphene nanoribbons (GNRs) and show that they can function as beam splitters or mirrors. These features are identified for electrons in the low-energy region where a single valence or conduction band is present. Our modeling is based on $p_z$ orbital tight binding with Slater-Koster-type matrix elements fitted to accurately reproduce the low-energy bands from density functional theory calculations. We analyze systematically all devices that can be constructed with either zigzag or armchair GNRs in AA and AB stackings. From Green's function theory the elastic electron transport properties are quantified as a function of the ribbon width. We find that devices composed of relatively narrow zigzag GNRs and AA-stacked armchair GNRs are the most interesting candidates to realize electron beam splitters with a close to 50:50 ratio in the two outgoing terminals. Structures with wider ribbons instead provide electron mirrors, where the electron wave is mostly transferred into the outgoing terminal of the other ribbon, or electron filters where the scattering depends sensitively on the wavelength of the propagating electron. We also test the robustness of these transport properties against variations in the intersection angle, stacking pattern, lattice deformation (uniaxial strain), inter-GNR separation, and electrostatic potential differences between the layers. These generic features show that GNRs are interesting basic components to construct electronic quantum optical setups.

Figure 1

Illustration of the general setup. (a) A four-terminal device is formed by two crossed 8-ZGNRs with a relative angle $\theta ={60}^{\circ }$. The bottom (top) ribbon is drawn in blue (red) with carbon atoms at each vertex. The four semi-infinite leads, numbered 1–4, are attached in the contact regions represented with red rectangles. The ribbons lie out of plane separated by a distance $d$ along the $z$ axis (see side view). Definition of the width $W$ of (b) ZGNRs and (c) AGNR in terms of the number of carbon atoms $N$ across the ribbon. The interatomic distance is denoted by $a$.

Figure 2

Geometries of the different stackings that can be constructed from the crossing of two GNRs with a relative angle of ${60}^{\circ }$. The bottom (top) ribbon is drawn in blue (red) with carbon atoms at each vertex. For ZGNR-based devices there exists only one AA- and one AB-stacked configuration, labeled (a) AB and (b) AA (exemplified here by 8-ZGNR). For AGNR-based devices there exist two AA- and two AB-stacked configurations, labeled (c) AA-1, (d) AA-2, (e) AB-1, and (f) AB-2 (exemplified here by 11-AGNR). The dashed lines show the symmetry (reflection) planes that preserve the Hamiltonian of each crossing when such operation is applied to them.

Figure 3

Spectral DOS of scattering electrons incoming from electrode $\alpha =1$ obtained from Eq. (9), for the specific geometries defined in Fig. 2: (a) 8-ZGNR AB, (b) 8-ZGNR AA, (c) 11-AGNR AA-1, (d) 11-AGNR AA-2, (e) 11-AGNR AB-1, and (f) 11-AGNR AB-2. The spectral DOS were calculated at $E=200$ meV for ZGNRs and at $E=0$ meV for AGNRs.

Figure 4

Full transmission probability matrix $T_{\alpha \beta }$ between all the electrode pairs for ZGNRs crossed in the AB configuration as a function of the ribbon width $W$ and electron energy $E$. Only data for the first subband are shown (white regions correspond to multiple electronic bands in the ribbons).

Figure 5

Figure of merit (FM) for systems composed of ZGNRs in (a) AB or (b) AA configurations. Black and red regions correspond to situations where a given device is suitable as a BS or M, respectively. White regions are unsuitable as BS or M because of large transmission into the other but the desired output ports.

Figure 6

Figure of merit (FM) for systems composed of AGNRs of the $3p+2$ family with (a) AA-1, (b) AA-2, (c) AB-1, and (d) AB-2 configurations. Black and red regions correspond to situations where a given device is suitable as a BS or M, respectively. White regions are unsuitable as BS or M because of large transmission into the other but the desired output ports.

Figure 7

Sketch of the geometrical distortions applied to the AB-stacked 8-ZGNR device. (a) Rotation by some small angle $\delta \theta$ around the configuration with a relative angle of $\theta ={60}^{\circ }$. The rotation is performed around the center of the scattering region, indicated by a black dot. (b) Translation of the on-top ribbon with respect to the lower one by an amount ${\mathrm{\Delta }}_x$ along the $x$ axis. (c) Strain $\varepsilon$ is applied along the periodic direction of each ribbon while keeping the center of the scattering (black dot) region unchanged.

Figure 8

Variation with respect to the rotation angle between two AB-stacked 8-ZGNRs. Reflection and transmission probabilities, (a) $R_1$ (b) $T_{12}$, (c) $T_{13}$, and (d) $T_{14}$, and (e) figure of merit as a function of the incoming electron energy $E-E_F$, obtained for different relative angles ($\theta$) between the ribbons (color lines). The reference probabilities ($\theta ={60}^{\circ }$) are plotted in black solid lines.

Figure 9

Variation with respect to the relative lateral displacement between two AB-stacked 8-ZGNRs. Reflection and transmission probabilities, (a) $R_1$ (b) $T_{12}$, (c) $T_{13}$, and (d) $T_{14}$, and (e) figure of merit as a function of electron energy $E-E_F$, obtained for different translation distances along the $x$ axis (${\mathrm{\Delta }}_x$) of the on-top ribbon (color lines). The reference probabilities (${\mathrm{\Delta }}_x=0$) are plotted in black solid lines.

Figure 10

Variation with respect to the applied uniaxial strain $\varepsilon$ along the periodic direction of each GNR for the two AB-stacked 8-ZGNRs. Reflection and transmission probabilities, (a) $R_1$ (b) $T_{12}$, (c) $T_{13}$, and (d) $T_{14}$, and (e) figure of merit as a function of electron energy $E-E_F$, obtained for different uniaxial strain $\varepsilon$ applied to both GNRs along the nonconfined direction (color lines). The reference probabilities ($\varepsilon =0$) are plotted in black solid lines.

Figure 11

Variation with respect to the inter-GNR separation of two AB-stacked 8-ZGNRs. Reflection and transmission probabilities, (a) $R_1$ (b) $T_{12}$, (c) $T_{13}$, and (d) $T_{14}$, and (e) figure of merit as a function of electron energy $E-E_F$ and inter-GNR separation $d$ (color lines). The reference probabilities ($d=3.34$ Å) are plotted in black solid lines.

Figure 12

Variation with respect to potential differences $V$ between the two AB-stacked 8-ZGNRs. Reflection and transmission probabilities, (a) $R_1$ (b) $T_{12}$, (c) $T_{13}$, and (d) $T_{14}$, and (e) figure of merit as a function of electron energy $E-E_F$, obtained for different values of $V$ (colored lines). The reference probabilities ($V=0$) are plotted in black solid lines.

Figure 13

Band structure of AB-stacked bilayer graphene along the $\mathrm{\Gamma }-K-M-\mathrm{\Gamma }$ path of the Brillouin zone, obtained with DFT (black and gray solid lines) and TB methods (red dashed lines), with the fitted hopping parameters described in the text. The bond length is set to $a=1.42$ Å and the interlayer separation to $d=3.34$ Å. Black lines correspond to the graphene $\pi$ bands (formed by the $p_z$ orbitals) while the gray lines show the graphene $\sigma$ bands absent in the TB model.

Figure 14

Reflection and transmission probabilities $R_1$ (black), $T_{12}$ (blue), $T_{13}$ (green), and $T_{14}$ (red), obtained with both TB (solid lines) and DFT (dotted lines) methods, through the devices of Fig. 2: two crossed 8-ZGNRs in configuration (a) AB and (b) AA, and two crossed 11-AGNRs in configuration (c) AA-1, (d) AA-2, (e) AB-1, and (f) AB-2.