## 2 Some Attenuation Factors of Standard Molecular Junctions  
In this review, we will consider specifically the non-resonant tunneling regime, which deals with small molecular lengths in the junction (typically below 5 nm [7]) and also low applied bias. In this case, the evolution of the conductance with respect to the molecular length can be written in the following form:  
\[\mathrm{G=A\ exp\ (-\beta L)}, \tag{1}\]  
where L is the molecular length, A is the pre-exponential factor representative of the contact resistance at the molecule-metal interface, and \(\beta\) is the attenuation factor [10]. This attenuation factor is representative of the current attenuation within the molecular junction, with respect to the molecular length. In order to redefine the context, the attenuation factor is zero for a pure metallic junction, as no attenuation occurs, whereas it is of the order of 25 nm\({}^{-1}\) for the vacuum where the current is fully attenuated. A standard example of attenuation factor measurement is alkane-based molecular junctions. Many studies have been devoted to this system, and the attenuation factor currently lies around 7-9 nm\({}^{-1}\)[10; 11; 12]. It has to be noted that this value is strongly dependent on the chemical nature of the molecular wire, since substituting carbon by silicon to constitute what is called oligosilane chains reduces the attenuation factor to 3.9 nm\({}^{-1}\)[13]. This effect is even more enhanced considering germanium-based molecular wire, to form a germane chain, leading to \(\beta\sim\) 3.6 nm\({}^{-1}\)[14]. In this respect, we can speculate that going down the lines of the periodic table, i.e., increasing the atomic number of the chain constituents, would lead to an important reduction in the attenuation factor--in other words, a higher conductance, potentially related to the increasing number of electrons per atom. On the other hand, still considering non-conjugated molecular chains, the opposite behavior is found in siloxane chains, where the repetition unit is based on a Si-O dimer, with a very important attenuation factor of \(\beta\)\(\sim\) 12.3 nm\({}^{-1}\)[15]--in other words, almost no conductance. Beyond the chemical nature of the molecular chain elements, the nature of the molecular bonding plays an important role. For example, coming back to carbon elements, aromatic chains present reduced attenuation with respect to alkane chains, with \(\beta\sim\) 2.5 nm\({}^{-1}\)[16]. Similarly, carotenoid polyenes present \(\beta\) of 2.2 nm\({}^{-1}\)[17] and oligothiophenes have \(\beta\) between 2 and 3 nm\({}^{-1}\)[18]. Consequently, one can deduce that the aromaticity of the molecular chain helps in reducing the attenuation factor; however, it is more or less always in the same range. Besides the above, some more complex molecular systems also exhibit even lower attenuation factors, around 10 times lower than the polyenes discussed previously. For example, polymethine dyes, which are \(\pi\)-conjugated compounds with an odd number of carbons, have attenuation factor \(\beta\sim\) 0.4 nm\({}^{-1}\)[19]. This is mainly attributed to greater electronic delocalization, in comparison to standard polyenes, enhanced by a smaller degree of bond order alternation. However, this behavior seems to be limited by the molecular length, leading to a more resistive junction for long molecular chains. In addition, the more exotic case of porphyrin polymers is very interesting. Indeed, when considering oligo-porphyrin molecular junctions, based on Zn porphyrins with pyridine ligands, one can observe a similar reduced attenuation factor \(\beta\sim\) 0.4 nm\({}^{-1}\)[20]. Actually, it seems that the molecular conductance has a strong temperature dependence and a weak length dependence, even though it does not correspond to the hopping mechanism described in the introduction. It is rather consistent with phase-coherent tunneling through the whole molecular junction.