## 2 Some Attenuation Factors of Standard Molecular Junctions
Even more surprising is the result obtained by [PERSON] et al., who have studied molecular chains of fused porphyrins [21]. Namely, in the considered porphyrin oligomers, the monomers connect directly with their nearest neighbor through the porphyrin cycle. In this case, contrary to the previous cases discussed here, the conductance increases with the distance, by more than a factor of 10, when a small bias (\(\sim\)0.7 V) is applied to the junction. This exceptional behavior is due to the evolution of the HOMO-LUMO gap that rapidly decreases with the molecular length, which compensates for the increased tunneling distance. Finally, the last example to be cited with respect to low attenuation factors is the work conducted by [PERSON] et al., who show a structural control on the electronic transport resonance in HS(CH\({}_{2}\))n[1,4 \(-\)C\({}_{6}\)H\({}_{4}\)](CH\({}_{2}\))nSH (\(n=\) 1, 3, 4, 6) metal-molecule-metal junctions, leading to very small attenuation factors [22]. This work offers very promising perspectives in gating the transport resonance in order to modulate the molecular junction behavior. In this discussion, we have mainly considered the case of single molecule junctions. An interesting question that arises is this: what happens if we consider not only a single molecule junction but also a large molecular area? In this case, several experiments have found that the attenuation factor \(\beta\) mainly remains the same, and only the prefactor of the conductance is modified, as for molecular junctions based on biphenyl, nitrophenyl, ethynyl benzene, anthraquinone, etc., where one can observe attenuation factors around 2.7 nm\({}^{-1}\)[23] (see Figure 1). As an important consequence, these results show that the study of single molecule junctions is not only important at the fundamental level. The understanding of the transport mechanism at the single molecule level and, in particular, the determination of the attenuation factor may be extrapolated to large molecular areas, which are currently used in molecular electronics devices. Another important aspect that can be stressed is that, below 5 nm, it seems that there is no significant difference in the attenuation factor with respect to the molecular backbone of the aromatic molecules considered in the junction. This might be attributed to the relative positions of the molecular electronic levels, which remain constantly pinned near the Fermi level at around 1.3 eV [23; 24]. We will see in the next sections that this position of the molecular levels is an important parameter to determine the attenuation factor. However, it has to be noted also that most of these experimental results have been obtained using molecular junctions in symmetric configurations, namely with the same electrode at each side of the junction (most of the time, metallic electrodes made of noble metals) and similar anchoring groups. Very few studies have considered non-symmetric junctions--for example, with different anchoring groups at each molecular end--which might have an influence on the position of the molecular levels and yield other kinds of information about the transport mechanism [25; 26]. In addition, as we will detail later, some recent experiments have started to use different electrodes at each end as well as  
Figure 1: Overlay of attenuation plots for eight different molecules constructed from J–V curves with different thicknesses of each structure (the length of the error bars is two standard deviations). The lines are least squares regression lines for aromatic (2.7 nm\({}^{-1}\)) molecules. The different abbreviations stand for the aromatic molecular names: NP =nitrophenyl, AQ = anthraquinone, NAB = nitroazobenzene, BrB = bromophenyl, EB = ethynylbenzene, AB = azobenzene and BTB = bisthenylbenzene. Adapted figure from [23] and with permission. Copyright (2010) American Chemical Society. non-metallic electrodes. Obviously, the question that arises is this: what is the fundamental element in a molecular junction that determines the attenuation factor?