# A Simple Model of Ballistic Conduction in Multi-Lead Molecular Devices
## 3 Multi-Lead Devices
### Symmetric Multi-Lead Devices  
The leads in symmetric devices have identical lead parameters, \(\beta_{\mathrm{L}}\), and identical connection parameters, \(\beta_{\mathrm{LM}}\). This enables a relatively easy simplification of the transmission formula (39) into a cluster expansion, involving pairs, triples, etc. of leads. All leads satisfy a single dispersion relation  
\[E=2\beta_{\mathrm{L}}\sin q_{\mathrm{L}}, \tag{41}\]and furthermore, the reduced lead matrix of Equation (31) is now  
\[\overline{\mathbf{p}}_{\mathrm{L}}=\beta_{\mathrm{L}}e^{-iq_{L}}\mathbf{1}_{l}- \beta_{\mathrm{LM}}^{2}\overline{\mathbf{U}}(E\mathbf{1}_{n}-\mathbf{\epsilon}_{ \mathrm{M}})^{-1}\overline{\mathbf{U}}^{\dagger}, \tag{42}\]  
where the bar over the matrix \(\overline{\mathbf{U}}\) indicates that we require rows of the matrix only for those atoms connected to leads. The rectangular matrix \(\overline{\mathbf{U}}\) is obviously not unitary and indeed gives different products  
\[\overline{\mathbf{U}}\overline{\mathbf{U}}^{\dagger}=\mathbf{1}_{l},\quad \text{but}\quad\overline{\mathbf{U}}^{\dagger}\overline{\mathbf{U}}\
eq\mathbf{ 1}_{n}, \tag{43}\]  
but the first of these and the diagonal nature of the first term in the \(l\)-dimensional matrix \(\overline{\mathbf{P}}_{\mathrm{L}}\), allow us to write  
\[\overline{\mathbf{P}}_{\mathrm{L}}=\overline{\mathbf{U}}\mathbf{Q}_{\mathrm{ M}}\overline{\mathbf{U}}^{\dagger}, \tag{44}\]  
where the _diagonal_\(n\)-_dimensional_ matrix, \(\mathbf{Q}_{\mathrm{M}}\), which involves all eigenstates, is  
\[\mathbf{Q}_{\mathrm{M}}=\beta_{\mathrm{L}}e^{-iq_{L}}\mathbf{1}_{n}-\beta_{ \mathrm{LM}}^{2}(E\mathbf{1}_{n}-\mathbf{\epsilon}_{\mathrm{M}})^{-1}. \tag{45}\]  
Equation (44) expresses the reduced lead matrix, \(\overline{\mathbf{P}}_{\mathrm{L}}\), as a triple product involving rectangular matrices. The Cauchy-Binet theorem [76; 77] can be used to expand the determinant of a product of two rectangular matrices. Hence, we obtain a further reduction in the quantities in Equation (33) as a sum over a set of products of \(l\)-dimensional principal minors  
\[\mathrm{D}_{\mathrm{L}}=\det\overline{\mathbf{P}}_{\mathrm{L}}=\sum_{\mathrm{ K}}\det\overline{\mathbf{U}}_{\mathrm{K}}\det(\mathbf{Q}_{\mathrm{M}}\overline{ \mathbf{U}}^{\dagger})_{\mathrm{K}}=\sum_{\mathrm{K}}\det\overline{\mathbf{U} }_{\mathrm{K}}\prod_{\mathrm{K}\in\mathrm{K}}(\mathrm{Q}_{\mathrm{M}})_{kk} \det(\overline{\mathbf{U}}^{\dagger})_{\mathrm{K}}, \tag{46}\]  
where \(\mathrm{K}=\{k_{1},k_{2},\cdots,k_{l}\}\) is an ordered subset of \(1,2,\cdots,n\), with \(|\mathrm{K}|=l\), and \(1\leq k_{1}<k_{2}\cdots<k_{l}\leq n\). By deriving this equation, we are able to separate the roles of eigenvectors, \(\overline{\mathbf{U}}\), and the state energies contributing to \(\mathbf{Q}_{\mathrm{M}}\). Further manipulation of Equation (46) uses algebra familiar in the reduction of matrix elements over Slater determinants in quantum chemistry. We can replace the restricted summation over the ordered sets, \(\mathrm{K}\), by unrestricted summations from \(1\) to \(n\) over the state indices (i.e., Huckel MO or GF pole indices), \(k_{1},\cdots,k_{l}\). Equalities between the \(k_{i}\) indices are allowed because they give rise to zero contributions in Equation (46), since they produce determinants, \(\det\overline{\mathbf{U}}_{\mathrm{K}}\), with identical columns. We can also use the antisymmetry of the determinants to replace the first determinant by a simple product. Hence, we can conclude that  
\[\mathrm{D}_{\mathrm{L}}=\sum_{k_{1},\cdots,k_{l}}\left(\prod_{p=1}^{l} \overline{\mathbf{U}}_{\mathrm{pk}_{p}}(\mathbf{Q}_{\mathrm{M}})_{k_{p}k_{p}} \right)\det(\overline{\mathbf{U}}^{\dagger})_{\mathrm{K}}, \tag{47}\]  
The next step is to expand the product of the \((\mathrm{Q}_{\mathrm{M}})_{k_{p}k_{p}}\) using Equation (45) to generate a cluster expansion in powers of \(\beta_{L}\) and \(\beta_{\mathrm{LM}}^{2}\). Whilst it is easy to pursue this generally, it is probably more enlightening to consider the example of a three-lead case explicitly.