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Landauer Formula

We consider the transport of electrons through a molecular system by modeling it as a one electron elastic scattering problem. The molecule acts as a defect between two metallic reservoirs of electrons. An electron incident from the source lead with an energy E, has a transmission probability T(E)to scatter through the molecule into the drain lead. A schematic of the model system is shown in Fig. 2.1. By determining the transmission probability for a range of energies around the Fermi energy $\epsilon _F$ of the source lead, the finite temperature, finite voltage, Landauer formula can be used to calculate the transmitted current I as a function of the bias voltage, V, applied between the source (left lead) and drain (right lead)

 \begin{displaymath}I(V) = \frac{2e}{h} \int_{-\infty}^{\infty}
dE\:T(E)\left( \f...
...E-\mu_{s})/kT] + 1} -
\frac{1}{\exp[(E-\mu_{d})/kT]+1} \right)
\end{displaymath} (1)

The two electro-chemical potentials $\mu_{s}$ and $\mu_{d}$, refer to the source and drain, respectively. They are defined to be, $\mu_{s} = \epsilon_{F} + eV/2$ and $\mu_{d} =
\epsilon_{F} - eV/2$. The differential conductance is then given by the derivative of the current with respect to voltage.


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