## 4 A Test Case: Alkane Chains
### Role of the Electrodes: Hybrid Junctions Using a Graphene Electrode
(**Right**) Evolution of the conductance as a function of the molecular length: experimental measurements on asymmetric metal–graphene junction (black), experimental measurements on symmetric metal–metal junction (blue) and conductance calculation on asymmetric metal–graphene junction (red). Reprinted figure from [40] and with permission. Copyright (2016) American Chemical Society. remains far from the Fermi level, at around 2 eV, as discussed previously, yielding an important electronic barrier that is reflected in a high attenuation factor (\(\beta\sim 8.6\) nm-1 or 0.8 per C atom [1,11]). Note that [PERSON] et al. have also studied the case of symmetric junctions, where it was argued that dipoles in symmetric junctions can result in the HOMO level being dragged down in energy [22]. If the second electrode is now a graphene plane, the interface dipole at the graphene side ruled by van der Waals interaction is much weaker than the one at the gold side, which is not cancelled in this situation. This results in an important charge transfer at the gold-molecule interface and a shift of the HOMO level toward the Fermi level. Consequently, the energy difference E\({}_{\text{Fermi}}\)-E\({}_{\text{HOMO}}\) is reduced to 0.4 eV, which reduces considerably the attenuation factor, to 0.4 per C atom, as observed experimentally. In this case, as we can see, the simple Simmons model is perfectly able to describe the electronic behavior of the junction. Moreover, we have observed that the use of a graphene electrode at one extremity breaks the electrostatic symmetry of the system, leading to a reduced attenuation factor for thiol groups, as compared to the same junction and two gold electrodes. Now, in order to extend the comparison to symmetric junctions, and to probe the theoretical model used for the thiol case, we will consider another hybrid junction with different anchoring groups, namely amine groups [46]. The same procedure has been applied here with I(s) measurements and DFT based electronic transport calculations. The corresponding results are presented in Figure 5. First, we consider the evolution of the conductance as a function of the molecular length, in a logarithmic scale. Similarly to the case of the thiol Au-graphene junction, an attenuation factor of 0.4 per C atom is also found in the amine case. This behavior is consistent with what happens with Au-Au junctions, as we have seen in the previous subsection. Indeed, for all the anchoring groups, the attenuation factor remains practically the same. Here, the introduction of a graphene electrode seems to result in a similar effect, but it is reduced to around half its value with respect to Au-Au junctions. What is more surprising here is the direct comparison of the conductance with the Au-Au junction and the same amine groups. Indeed, while, for the thiol case, we observed greater conductance above a certain molecular length due to the reduced attenuation factor, we can observe for the amine group a more important conductance independently of the molecular length is, as shown in Figure 5. Even more surprisingly, while, for the Au-Au junction, the thiol case presents higher conductance than the amine one (due to better contact and lower contact resistance), in the Au-graphene junction, the amine case presents higher conductance than the thiol one. Figure 5: (**Left**) Evolution of the conductance as a function of the molecular length for molecular junctions with different anchoring groups and electrodes: conductance calculation on Au–graphene junctions with amine group (green), conductance measurements on Au–graphene junctions with amine group (black), conductance measurements on Au–graphene junctions with thiol group (red) and conductance measurements on Au–Au junctions with amine group (blue). (**Right**) Density functional  
theory (DFT) calculated density of states (DOS) for Au-graphene and Au-Au molecular junctions with amine anchoring groups. Reprinted figure from [46] and with permission. Copyright (2017) American Chemical Society.