# A Simple Model of Ballistic Conduction in Multi-Lead Molecular Devices
## 6 Results
### Devices Based on the Benzene Ring  
As a first example, we take the six-membered ring of carbon atoms, which can be connected as a two-lead device in three ways ('ortho','meta', 'para') and as a three-lead device in six ways, leading to the nine transmission curves shown in Figure 4. All curves are symmetric about the Fermi energy in the Huckel model, as the six-ring is a bipartite graph. Transmission curves for the three two-lead devices are qualitatively different. The ortho and para devices have non-zero transmission at the Fermi energy, with different patterns in the wings, as predicted by the analytical expression for \(T(E)\) of a cycle [55] and selection rules [61; 63] for conduction at degenerate and non-degenerate eigenvalues (here, \(\alpha\pm\beta\)). In contrast, the meta device is insulating at the Fermi level.  
Plots for the various three-lead devices show a variety of patterns, but it is striking that the overall shapes of the curves can be interpreted in terms of constituent two-lead components: to a first approximation, the curve \(T(E)\) for the three-lead device with source \(1\), and sinks \(a\) and \(b\) has lead contributions that follow the patterns for devices \((1,a)\) and \((1,b)\) in the region of the Fermi level but are damped in the higher and lower energy wings. The implication is that the interference terms (see Equation (60)) have smooth, predictable effects on \(T(E)\) for the composite device. This pattern is especially clear for the symmetric \((1,3,5)\) device (Figure 4i), where the source is in a meta relations to both sinks, and the Fermi-level transmission is zero. In fact, this device is a CBSD: interference terms between leads do not appear in the numerators of the lead currents, and the device is interference-free. In other devices such as \((1,2,3)\) and \((1,3,4)\), a vanishing Fermi-level contribution from the meta constituent device is masked by the contribution from the other constituent. The counting of ortho, meta and para constituent devices works well as a rough guide to \(T(E)\), though we may expect this simplicity to be diluted in larger systems and devices with more leads.  
The final row of Figure 4 shows the transmission curves calculated at the GF2 level for \((1,2)\), \((1,4)\) constituent devices, flanking the plot for the composite three-lead device \((1,2,4)\). The plots show strong resemblances to those calculated at the Huckel level, and interpretation of the lead contributions in terms of constituent devices survives intact in the correlated treatment.  
Figure 3: Numbering schemes for the molecules used here in examples of multi-lead devices. They are (**a**) benzene, (**b**) anthracene, (**c**) pentalene, and(**d**) heptatriene. The gold pseudo-atom symbol shows the position of the source lead; the positions of sink leads are given by the device specifications in the text. Green pseudo-atoms in (**d**) show the sink leads for two possible CBSD.  
Figure 4: Transmission curves for multi-lead devices based on benzene: Hückel theory results for **(a)** 1,2-; **(b)** 1,3-; **(c)** 1,4-; **(d)** 1,2,3-; **(e)** 1,2,4-; **(f)** 1,2,5-; **(g)** 1,2,6-; **(h)** 1,3,4-; and **(i)** 1,3,5-attachments of leads; GF2 results for **(j)** 1,2,-; **(k)** 1,2,4-; and **(l)** 1,4-devices. Lead 1 is the source, with transmission equal to the total envelope, shown in black. Transmissions for the second and third leads (counting clockwise in the numbering scheme of Figure 3) are shown in red and green, respectively. In the case of repetition of identical curves, later colours overwrite earlier ones. Red vertical lines show positions of ionisation poles, whilst blue lines mark attachment pole positions (in either Hückel or second-order correlated GF calculations). Dotted vertical lines mark shake poles for the correlated calculations. Lead and connection parameters are \(\beta_{\rm L}=2.0\) and \(\beta_{\rm LM}=1\).