To model the periodicity in the graphene and interaction between the molecules in the SAM, the unit cell was repeated using a Bravias lattice with 30 k-points in the y direction. This models an SAM where molecules are ~2 nm apart. The gold electrode is considered to be a nanowire and not periodic. A mean-field Hamiltonian and an overlap matrix were extracted from this converged calculation.

To model the source-drain and VG in the experiment, we used a model where the gold lead is earthed, and therefore, the gold Fermi energy Embedded Image is not affected by the source-drain or gate voltage. However, the Fermi energy of graphene is a function of the source-drain and gate voltages via the equationEmbedded Image(1)where VD and VG are the source-drain and gate voltages, and α and β are the experimental lever arms, which could vary in each experiment. Similarly, applying a VG can move the energy levels of the molecule up and down in energy (Eqs. 2 and 3)Embedded Image(2)Embedded Image(3)

The value for current is given by Eq. 4Embedded Image(4)where T(E, VD, VG), is the transmission coefficient from lead 1 to lead 3 calculated using quantum transport code the GOLLUM (26). In what follows, γG = 1. When comparing theory with experiment, e.g., in Fig. 2, a lever arm of α = 0.44 was used.

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