## 4 A Test Case: Alkane Chains
### Role of the Electrodes: Hybrid Junctions Using a Graphene Electrode  
Most of the conductance measurements in molecular electronics have used metallic electrodes and, in particular, noble metals, either using mechanically controlled break junctions (MCBJ) [39], scanning tunneling microscopy (STM) [40], conductive probe atomic force microscopy (CP-AFM) [41] or also the STM-based I(s) method developed by [PERSON] et al. [42]. Nevertheless, an important number of recent works have been devoted to the use of carbon electrodes or even graphene substrates [43; 44]. In this respect, we will discuss here the attenuation factors measured experimentally and determined theoretically on hybrid molecular junctions with a gold electrode at one molecular end and a graphene electrode at the other. We will also consider the role of different anchoring groups using this graphene electrode. First, we start by considering alkanedithiol molecules of different lengths, probed experimentally using the I(s) method and modeled using DFT and electronic transport calculations to interpret the obtained results [45]. An atomic model of the system is represented in the left part of Figure 4 for a butanedithiol sandwiched between a gold and a graphene electrode, forming a hybridmetal/molecule/graphene junction. The conductance has been measured for different molecular lengths between 2 and 12 -CH\({}_{2}\) groups, allowing us to deduce the length dependence of the conductance. In this respect, the use of a graphene electrode does not affect the exponential conductance decay. In parallel, electronic transmission spectra and conductance have been calculated after DFT structural optimization of the junctions and density of states calculations. In addition, the contact resistance has been determined to be significantly larger than what is obtained for a standard symmetric junction with two gold electrodes. This is due to the weak coupling (of van der Waals nature) at the graphene-molecule interface. The length evolution of the experimental and theoretical conductance is represented on the right part of Figure 4.  
As a result, the attenuation factor for this hybrid junction is approximately half of the one obtained for its symmetric metal-molecule-metal counterpart. Namely, the attenuation factor now lies at around 0.4 per carbon atom. Hence, even though the contact resistance is quite high for the hybrid junction due to the weakly coupled interface with graphene, for molecules longer than ~1 nm, the metal-molecule-graphene junction turns to be more conductive than the standard metal-molecule-metal one. Moreover, for similar molecular lengths, the hybrid junction is also longer than the standard one, due to the van der Waals contact which requires a 3 A distance between the molecule and graphene. Therefore, the question arises: how can such unusual behavior occur? We will try to understand it in the frame of the Simmons model, the most common theoretical approach to determine an attenuation factor that we have discussed in Section 3. Hence, the electronic behavior can be explained as follows by the different couplings at the electrode-molecule interfaces and the different molecular level alignments. When a thiol-terminated alkane is adsorbed on a gold surface, due to the strong interface electrostatic dipole, there is an important charge transfer, which partially depopulates the molecule and moves the HOMO level toward the Fermi level of the surface. If the same molecule is contacted between two gold electrodes, there is an interface dipole at each molecular extremity, in opposite directions, leading to a cancellation of the two dipoles. Then, the HOMO level  
Figure 4: (**Left**) Schematic representation of the hybrid metal–molecule junction formed in the experiment for conductance measurements.