# A Simple Model of Ballistic Conduction in Multi-Lead Molecular Devices
## 2 Two-Lead Devices  
In the standard two-lead SSP model, the device consists of a central molecule that is attached by single-atom contacts to the leads (Figure 1). In the steady state, the source lead supports an incoming electron wave that undergoes partial reflection at the interface with the molecule, and the transmitted part of the wave scatters through the molecule and exits into the sink lead. The essential observation behind the model [40] is that the two semi-infinite leads may be replaced by source and sink pseudo-atoms equipped with appropriate complex potentials, which play the roles of delivering the transmitted fraction of the electron to the molecule and removing it downstream, respectively. In the purely graph theoretical version of the model, this step allows replacement of a doubly infinite system by one in which the \(n\times n\) Hermitian (in fact, real symmetric) molecular Hamiltonian matrix is expanded to an \((n+2)\times(n+2)\) non-Hermitian matrix. It is convenient to recast this generalised eigenvalue problem in the form of an inhomogeneous linear equation [63]:  
\[\mathbf{Pc}=\mathbf{b}, \tag{1}\]  
where \(\mathbf{P}\) is the SSP matrix, the solution vector \(\mathbf{c}\) gives the device wavefunction specified by complex entries on the \(n+2\) vertices, and \(\mathbf{b}\) is the inhomogeneity term that embodies the boundary conditions. The solution to this equation for each possible value of the energy of the incoming electron yields a compact expression for the zero-voltage transmission as a function of energy, \(T(E)\), and as a bonus, gives a useful analysis of current in terms of bond or molecular-orbital (MO) contributions. The transmission function can be cast in terms of the characteristic polynomials of four graphs, \(G\), \(G-\overline{\Gamma}\), \(G-\overline{\mathbb{R}}\), and \(G-\overline{\Gamma}-\overline{\mathbb{R}}\), where \(\overline{\Gamma}\) and \(\overline{\mathbb{R}}\) are the centres within the graph that are in direct contact with the leads. These polynomials are denoted \(s\), \(t\), \(u\), and \(v\) in [55]. Details of the algebra are given in the reference, but the result is that the device transmission has the form [63]:  
\[T(E)=\mathcal{B}(q_{\mathrm{L}},\,q_{\mathrm{R}})\frac{j^{2}}{|D|^{2}}, \tag{2}\]  
where  
\[\mathcal{B}(q_{\mathrm{L}},\,q_{\mathrm{R}})=(2\beta_{\mathrm{L}}\sin q_{ \mathrm{L}})(2\beta_{\mathrm{R}}\sin q_{\mathrm{R}})\beta_{\mathrm{LM}}^{2} \beta_{\mathrm{RM}}^{2} \tag{3}\]  
is a band-pass function ensuring that the electron energy is within the conduction bands of the leads, and the denominator is  
\[D(E)=\beta_{\mathrm{L}}e^{-iq_{\mathrm{L}}}\beta_{\mathrm{R}}e^{-iq_{\mathrm{ R}}}s-\beta_{\mathrm{R}}e^{-iq_{\mathrm{R}}}\beta_{\mathrm{LM}}^{2}t-\beta_{ \mathrm{L}}e^{-iq_{\mathrm{L}}}\beta_{\mathrm{RM}}^{2}u+\beta_{\mathrm{LM}}^{ 2}\beta_{\mathrm{RM}}^{2}v, \tag{4}\]  
where the polynomials \(s\), \(t\), \(u\), and \(v\) are to be defined below in (7), The incoming and outgoing wavevectors \(q_{\mathrm{L}}\) and \(q_{\mathrm{R}}\) satisfy the Huckel dispersion relations for the leads:  
\[E=\alpha_{\mathrm{L}}+2\beta_{\mathrm{L}}\cos q_{\mathrm{L}}=\alpha_{\mathrm{ R}}+2\beta_{\mathrm{R}}\cos q_{\mathrm{R}}, \tag{5}\]  
and we take the symmetric case where \(\alpha_{\mathrm{L}}=\alpha_{\mathrm{R}}=\alpha\), and \(\beta_{\mathrm{L}}=\beta_{\mathrm{R}}\), and refer the zero of all energies to \(\alpha\). The polynomial \(j^{2}\) satisfies [74]:  
\[j^{2}=ut-sv, \tag{6}\]  
Figure 1: The SSP model of a molecular device. (**a**) The full system: a molecule attached to two semi-infinite leads via single-atom connections. Black and grey vertices are conjugated \(\pi\) centres within the molecular region; gold vertices are on the leads. (**b**) The SSP graph: Left (source) and right (sink) leads are replaced by pseudo-atoms \(\mathrm{L}\) and \(\mathrm{R}\) linked, respectively, to atoms \(\overline{\mathrm{L}}\) and \(\overline{\mathrm{R}}\) within the molecule. Black vertices are molecular centres directly attached to leads.