# A Simple Model of Ballistic Conduction in Multi-Lead Molecular Devices
## 3 Multi-Lead Devices
### Analysis of Interference in Multi-Lead Transmission  
In order to appreciate the nature of the transmission for multi-lead devices, we need to examine the numerator and denominator quantities \(|D_{\mathrm{L}\mathrm{P}1}|^{2}\) and \(|D_{\mathrm{L}}|^{2}\). Again taking the three-lead case as the example, we have  
\[|(D_{\mathrm{L}})_{\mathrm{P}1}|^{2}=\beta_{\mathrm{L}}^{2}\hat{k}_{\mathrm{L},\,\mathrm{P}}^{2}-E\hat{k}_{\mathrm{1},\,\mathrm{P}}\hat{k}_{\mathrm{1}q, \mathrm{P}q}+\hat{k}_{\mathrm{1}q,\mathrm{P}q}^{2}\,,\quad\text{ for }q\
eq\mathrm{P}\,. \tag{60}\]  
The first term in the expansion involves the same rank 1 off-diagonal SP that appears in the two-lead device constructed from leads 1 and \(\mathrm{p}\), albeit with a different \(|D_{\mathrm{L}}|^{2}\) denominator. We shall refer to this as the _direct_ term for the _constituent device_\((1,\mathrm{p})\). The remaining two terms represent the effect of the third lead (i.e., \(\mathrm{q}\)) on the conduction through the second lead (\(\mathrm{p}\)). We shall call these _mixed interference_ and _pure interference_ terms, respectively. Each term has a different dependence on the lead parameter, \(\beta_{\mathrm{L}}\), and the lead-molecule parameter, \(\beta_{\mathrm{L}M}\) (cf. Equation (9)). The direct term varies as \(\beta_{\mathrm{L}}^{2}\beta_{\mathrm{L}M}^{4}\), whilst mixed and pure interference components vary as \(\beta_{\mathrm{L}}\beta_{\mathrm{L}M}^{6}\) and \(\beta_{\mathrm{L}M}^{8}\), respectively. The different scaling implies that the relative importance of direct and interference effects may change markedly for different choices of the relative sizes of these parameters. The denominator term in the transmission is expressed using our trace notation as  
\[|D_{\mathrm{L}}|^{2} =\beta_{\mathrm{L}}^{6}+\beta_{\mathrm{L}}^{4}\Bigl{[}-E\,\hat{k} ^{(1)}+\Bigl{(}(\hat{k}^{(1)})^{2}-2\,\hat{k}^{(2)}\Bigr{)}\Bigr{]}+\beta_{ \mathrm{L}}^{2}\Bigl{[}E^{2}\,\hat{k}^{(2)}+E\,(-\hat{k}^{(1)}\,\hat{k}^{(2)}+ 3\,\hat{k}^{(3)})\] \[+\Bigl{(}(\hat{k}^{(2)})^{2}-2\,\hat{k}^{(1)}\,\hat{k}^{(3)} \Bigr{)}\Bigr{]}-\hat{k}^{(3)}\,\Bigl{[}E^{3}-E^{2}\,\hat{k}^{(1)}+E\,\hat{k}^{ (2)}-\hat{k}^{(3)}\Bigr{]} \tag{61}\]  
Equation (61) is derived for the symmetric three-lead device with \(\beta_{\mathrm{L}\mathrm{M}1}=\beta_{\mathrm{L}\mathrm{M}2}=\beta_{\mathrm{L }\mathrm{M}3}=\beta_{\mathrm{L}\mathrm{M}}\), and so the two-lead device cannot be recovered simply by taking \(\beta_{\mathrm{L}\mathrm{M}3}\to 0\), but in the limit that \(\hat{k}^{(3)}\to 0\), Equation (61) tends to a product of a factor \(\beta_{\mathrm{L}}^{2}\) multiplied by the denominator in Equation (15). The transmission formula for sink lead \(\mathrm{p}\) in multi-lead symmetric devices has the pre-factor \(N_{\mathrm{I}}^{2}N_{\mathrm{p}}^{2}=(4\beta_{\mathrm{L}}^{2}-E^{2})\), which gives the quadratic rise and fall in transmission as a function of energy near the lower and upper band edges, as in the two-lead device. The analysis of interference terms can be extended to the case where the device has more leads. In the case of \(l\) leads, the leading term in the numerator is of the form  
\[|(D_{\mathrm{L}})_{\mathrm{P}1}|^{2}=\beta_{\mathrm{L}}^{2l-4}\hat{k}_{ \mathrm{P}1}^{2}+\cdots\,. \tag{62}\]  
It has the same SP as the cognate constituent two-lead device, with a dependence on \(\beta_{\mathrm{L}\mathrm{M}}^{4}\), and this can be expected to be the most important term in the transmission spectrum. The other terms in \(|(D_{\mathrm{L}})_{\mathrm{P}1}|^{2}\) represent sink-lead interference terms. They form a cluster expansion with the inclusion of higher rank SPs up to the number of leads, \(l\), and traces over all sink leads other than \(\mathrm{p}\). An analysis in terms of a hierarchy of constituent two-lead, three-lead,  etc. devices could be envisaged. The equations for symmetric devices, exemplified by (60) and (61) can be used to derive general features of transmission spectra, selection rules for Fermi transmission, and systematic trends in interference effects.