## 4 A Test Case: Alkane Chains
### Role of the Electrodes: Hybrid Junctions Using a Graphene Electrode
In order to understand this particular behavior, we have calculated the density of states (DOS) for the alkane-based junction with amine anchoring groups, using Au-Au or Au-graphene electrodes. The result is represented in the right part of Figure 5.  
The calculated DOS allows us to determine the position of the HOMO level (which is the closest level to the Fermi level and therefore the conductive one in the junction, as for the thiol junction), so that we can apply the Simmons model used previously to determine the attenuation factor and the overall conductance. However, the calculated DOS indicates here a HOMO level located not around 0.4 eV below the Fermi level but 3.5 eV below the Fermi level for the amine anchoring group. As a consequence, the use of such a big potential barrier in the Simmons model would necessarily lead to a huge attenuation factor, which is not in agreement with the experimental observations. Therefore, this simple example highlights some limitations of the models presented in Section 3. Indeed, other models based on complex bandstructure calculations would not help either in determining correctly the attenuation factor, as electrode asymmetry cannot be taken into account. Note that the determination of the attenuation factor for the symmetric Au-Au junction is complicated as well, since, in that case, the energy barrier is about 5.5 eV, which does not fit into the Simmons model either. From this system, we can deduce that there is no universal model to calculate accurately the attenuation factors in molecular junctions. Hence, the Simmons model can be used only for specific ranges of electronic potential barriers, and the other models seem to be valid only for symmetric junctions, without properly considering the influence of the anchoring groups or the electrodes. In [46], this problem has been solved by considering that, between the Au-Au and the Au-graphene amine junctions, the HOMO level has been relocated from the same amount as in the Au-Au and the Au-graphene thiol junctions, namely around 2 eV (as indicated by the green arrow on Figure 5). Since the attenuation factors are the same for both anchoring groups in Au-Au junctions, it can be deduced that they should be the same for the Au-graphene junctions. The underlying reason is the fact that the coupling to the electrode and the molecular conduction are different for thiol and amine anchoring groups. The thiol groups connect very well to the electrodes, as of \(\pi\) symmetry, which explains the small electronic barrier, whereas the amine groups connect poorly to the electrodes, as of \(\sigma\) symmetry. Conversely, the thiol groups connect poorly to the alkane molecular backbone, yielding low intramolecular electronic propagation, whereas the amine groups connect much better, which increases the intramolecular propagation. One effect compensating the other, both anchoring groups lead to the same attenuation factor, mainly depending on the electrodes. This balance between coupling to the molecular backbone, leading to a good intramolecular conductance, and coupling to the electrode, which reduces the interface potential energy barrier, also explains the similar attenuation factors for different anchoring groups in alkane-based symmetric junctions between gold electrodes. One last point has to be clarified regarding the overall conductance of the amine junction, and the explanation is found again in the DOS calculation. Indeed, the HOMO level is different in the case of Au-graphene electrodes, due to a level splitting with respect to the Au-Au junction. Indeed, the introduction of the graphene electrode breaks the symmetry of the molecular junction (this effect was also observed for the thiol junction, where the symmetry breaking was seen in the non-compensation of the electric dipoles) and splits the original HOMO level, leading to a reduced molecular gap (by about 2 eV).