# A Simple Model of Ballistic Conduction in Multi-Lead Molecular Devices
## 8 Conclusions  
Working equations have been derived for an extended version of the SSP model for calculation of ballistic currents flowing through molecular conductors under a potential difference. They constitute the new Source-and-Multiple-Sink Potential model. The new model describes systems with an incoming electron travelling along a single source lead that drains out through the molecule into multiple sinks. As with its SSP predecessor, SMSP lends itself to a graph-theoretical approach, which leads to a formulation that is compatible with, but not confined to the Huckel/tight-binding description of electronic structure of \(\pi\) systems. SMSP therefore retains the advantages of SSP. It yields closed-form expressions for device transmission as a function of electron energy: for general lead parameters (Section 3.2); for chemically symmetric leads (Section 3.3); and for leads with coverage of all \(\pi\)-centres (Section 4.1), or of all starred or unstarred centres of an alternate \(\pi\)-system (Section 4.3). As its SSP predecessor did, the SMSP model gives predictions for qualitative selection rules and generic patterns of conduction for families of molecular systems, such as the opposite behaviour of aromatic and anti-aromatic rings at the Fermi level (Section 6.4). In addition to a calculus for the prediction of total transmission with energy, the model allows for an analysis of molecular current at various levels of detail: by bond current, by orbital-based molecular channels, and by destination lead (Section 5). Patterns of current in multi-lead devices can be interpreted as built up from notional constituent two-lead devices, where SMSP gives a natural breakdown into direct contributions, interference terms (again governed by selection rules), and mean-field corrections (Section 3.5). The wide-band limit yields particularly simple expressions for transmission involving only the characteristic polynomials of the set of vertex-deleted graphs (Section 3.4). In the hypothetical limit of full or half coverage of a molecule by leads, the specific interference effects are predicted to vanish (Sections 4.1 and 4.3). The symmetry of the device can also cause the vanishing of interference effects (Section 5.3) when the connection point for the sink leads are equivalent. SMSP, similar to SSP, is a parameterised model, with two parameters \(\beta_{\text{L}}\), and \(\beta_{\text{LM}}\) (defined in units of \(\beta\), the resonance integral for the molecule). Adjusting the parameters affects the detailed appearance of the transmission spectrum and the relative importance of interference effects. The effective lead parameter \(\beta_{\text{L}}\) is likely to be larger in magnitude than \(\beta\), even in all-carbon devices since the co-ordination number is higher with structured leads. The molecular energy levels, therefore, lie inside the band as assumed in our sample calculations. The role of \(\beta_{\text{LM}}\) is to describe the perturbation of the molecule by the leads, the effect of which is to move poles of the molecular GF off the real axis, and hence, lower \(\beta_{\text{LM}}\) values give sharper transmission peaks. Interference effects become more important as the ratio \(\beta_{\text{LM}}^{2}/\beta_{\text{L}}\) increases, as shown in (Section 3.5). Perhaps the main advantage of models such as SMSP is that many interpretative features persist when more realistic models of electronic structure are used. In particular, the analysis by molecular-orbital-based channels survives the introduction of electron-electron interaction into the theory. Thus far, we are not aware of experimental work on molecular systems with dense attachment of leads to single molecules. Mechanically controlled break-junction (MCB) techniques [83] provide IV characteristics that give insight into averaged conduction for two-lead molecular devices rather than their detailed transmission spectra.