## 4 A Test Case: Alkane Chains
### Role of the Anchoring Groups: Asymmetric Junctions  
As we know now, symmetry effects are very important in electronic transport in molecular junctions. In particular, we have seen that it has a strong influence on the attenuation factor. For this reason, it is interesting also to see what would be the influence of considering different anchoring groups at each molecular side to connect the electrodes to the molecules. In this respect, we highlight here the interesting case of hybrid Au-S-alkane-COOH-graphene molecular junctions, where thiol and carboxylic acid groups have been used at each extremity of the junction with gold and graphene electrodes [49]. An atomic representation of the molecular junctions studied here is represented in Figure 6.  
Obviously, from the theoretical considerations discussed above, there was no valid approach until now that was able to describe accurately such an asymmetric junction. Moreover, this type of junction has also been studied previously using Au-Au electrodes. As a reminder, the attenuation factor of junctions using only thiol or only carboxylic acid as anchoring groups are very similar, around 0.8-0.9 per C atom. Then, the corresponding attenuation factor for an asymmetric junction using different anchoring groups was found to be of the same magnitude, so without important change with respect to the symmetric junctions [25]. When considering junctions with Au-graphene electrodes, the situation is slightly different. Indeed, for a symmetric junction with carboxylic acid at both ends, the attenuation factor is around 0.7 per carbon atom, probably due to the low coupling of the carboxylic acid to the electrodes, which does not make a lot of difference in the situation of two gold electrodes. However,the situation is slightly different in the Au-graphene case, where the attenuation factor is around 0.4 per carbon atom, namely the same value as that found previously for thiol and amine groups in Au-graphene junctions. Interestingly, the electronic behavior is rather different. In thiol overall, but also in amine molecular junctions, the electronic transport is achieved through the HOMO level which is the closest level to the Fermi level. Here, as shown in Figure 7, the Fermi level is rather located near the middle of the HOMO and LUMO gap of the molecule, similarly to what is observed in the same junction with carboxylic acid at each molecular end. In terms of electronic transport, it means that there is no dominant molecular level (HOMO or LUMO) to the molecular conductance.  
From this result, we can deduce that molecular junctions with asymmetries in the anchoring groups see their electronic transport driven by the anchoring group that couples the most, even though the electronic structure is also strongly affected by the less coupled anchoring group. In other words, the current flows along the most conductive channel, similarly to what happens at the macroscopic  
Figure 6: Atomic representation of the different hybrid Au–graphene alkane-based molecular junctions using thiolate and carboxylic acid anchoring groups at each extremity. Reprinted figure from [49] and with permission. Copyright (2019) Wiley-VCH Verlag GmbH & Co.  
Figure 7: DFT calculated electronic transmissions for Au–graphene molecular junctions with carboxylic acid anchoring groups at each extremity (pink) and a thiol group at one extremity and a carboxylic acid at the other one (blue). Reprinted figure from [49] and with permission. Copyright (2019) Wiley-VCH Verlag GmbH & Co.  
scale, namely the channel which presents the best coupling at the molecule-electrode interface or the lowest contact resistance. Certainly, this constitutes only one particular study of asymmetric junctions in terms of anchoring groups and many other examples should be probed to extract trends that are more general. Nevertheless, it appears rather intuitive that the most conductive anchoring group will favor the most conductive channel and therefore the lowest attenuation factor.