# A Simple Model of Ballistic Conduction in Multi-Lead Molecular Devices
## 3 Multi-Lead Devices
### The SMSP Equations  
The Source-Multiple-Sink potential (SMSP) equations are modelled on our previous work with SSP [63], so that we can write them in the form (1) as  
\[\begin{pmatrix}\mathbf{P}_{\text{L}}&\mathbf{P}_{\text{LM}}\\ \mathbf{P}_{\text{ML}}&\mathbf{P}_{\text{M}}\end{pmatrix}\begin{pmatrix} \mathbf{\epsilon}_{\text{L}}\\ \mathbf{\epsilon}_{\text{M}}\end{pmatrix}=\begin{pmatrix}\mathbf{b}_{\text{L} }\\ \mathbf{0}\end{pmatrix}, \tag{19}\]  
where the \(l\)-dimensional (diagonal) lead block of the S(M)SP matrix, \(\mathbf{P}\), is  
\[(\text{P}_{\text{L}})_{\text{pq}}=\delta_{\text{pq}}\beta_{\text{L}_{\text{p}} }e^{-i\eta_{\text{L}_{\text{p}}}}=(\text{g}_{\text{L}}^{-1})_{\text{pp}}, \tag{20}\]  
Each nonzero element in \(\mathbf{P}_{\text{L}}\) is the matrix element for the terminal atom p of the inverse Green's function of the semi-infinite linear chain, \(\text{g}_{\text{L}}^{-1}\).  
The molecule block is the characteristic matrix constructed from the molecular adjacency matrix, \(\mathbf{A}_{\text{M}}\),  
\[\mathbf{P}_{\text{M}}=E\mathbf{1}-\mathbf{A}_{\text{M}}=\mathbf{U}(E\mathbf{1 }_{n}-\mathbf{\epsilon}_{\text{M}})\mathbf{U}^{\dagger}=\text{g}_{\text{M}}^ {-1}. \tag{21}\]  
The diagonal matrix, \(\mathbf{\epsilon}_{\text{M}}\), is the matrix of \(n\) molecular-orbital eigenvalues, and \(\mathbf{U}\) is the associated \(n\)-dimensional matrix of molecular-orbital eigenvectors. The unitarity of the eigenvectors implies that  
\[\mathbf{U}\mathbf{U}^{\dagger}=\mathbf{1}_{n}=\mathbf{U}^{\dagger}\mathbf{U}. \tag{22}\]  
Figure 2: The SMSP model of a molecular device. (**a**) The full system: a molecule attached to \(l\) semi-infinite leads via single-atom connections. Black and grey vertices are conjugated \(\pi\) centres within the molecular region; gold vertices are centres on the leads. (**b**) The SMSP graph: Each lead is replaced by a pseudo-atom with the index \(p=1,\ldots,l\), where lead 1 is the source and all others are sinks. Black vertices are molecular centres directly attached to leads and indexed correspondingly. Grey vertices are molecular vertices that are not directly linked to a lead and are indexed \(l+1,\ldots,n\).  
The lead-molecule block, \(\mathbf{P}_{\text{LM}}=\mathbf{P}_{\text{LM}}^{\dagger}\), is an \(l\times n\)-dimensional matrix with elements  
\[(\mathbf{P}_{\text{LM}})_{\text{Pq}}=-\delta_{\text{Pq}}\beta_{\text{LM}_{ \text{P}}}. \tag{23}\]  
On the right-hand side of Equation (19), the inhomogeneity vector, \(\mathbf{b}_{\text{L}}\), has a single non-zero element  
\[(\mathbf{b}_{\text{L}})_{\text{P}}=-iN_{\text{L}}\delta_{\text{P} 1}, \tag{24}\]  
which corresponds to the fact that lead 1 is the single source lead. The vector \(\mathbf{b}_{\text{L}}\) ensures that the system of equations satisfies the correct boundary conditions for a semi-infinite source lead with current travelling in the forward direction with pseudo-momentum, \(q_{\text{L}}\). The factor  
\[N_{1}=\sqrt{2\beta_{\text{L}_{1}}\sin q_{\text{L}_{1}}}, \tag{25}\]  
is the normalising factor (cf. [63]), ensuring that the forward current is equal to unity.  
The SMSP solution vector \(\mathbf{c}\) in Equation (19) is the (complex) device wavefunction. Consequently, we can write the transmission from the molecule to lead p using the standard formula for the electron current [2] as  
\[T_{\text{M}\to\text{P}}=2\,\beta_{\text{LM}_{\text{P}}}\Im\Big{(}c_{\text{ MP}}^{\ast}c_{\text{LP}}\Big{)}. \tag{26}\]  
This formula has the correct sign for transmission into sink leads (p \(\
eq\) 1), but the sign must of course be reversed to obtain the transmission from the source lead (p = 1).