Estimating the Number of Molecules in Molecular Junctions Merely Based on the Low Bias Tunneling Conductance at Variable Temperature

Temperature ($T$) dependent conductance $G = G(T)$ data measured in molecular junctions are routinely taken as evidence for a two-step hopping mechanism. The present paper emphasizes that this is not necessarily the case. A curve of $\ln G$ versus $1/T$ decreasing almost linearly (Arrhenius-like regime) and eventually switching to a nearly horizontal plateau (Sommerfeld regime), or possessing a slope gradually decreasing with increasing $1/T$ is fully compatible with a single-step tunneling mechanism. The results for the dependence of $G$ on $T$ presented include both analytical exact and accurate approximate formulas and numerical simulations. To be specific, we analyze in detail data available for molecular junctions based on ferrocene (Fc). As a particularly important finding, we show how the present analytic formulas for $G=G(T)$ can be utilized to compute the ratio $f = A_{\mbox{\small eff}} / A_n$ between the effective and nominal areas of large area Fc-based junctions with an EGaIn top electrode. Our estimate $f\approx 0.6 \times 10^{-4}$ is comparable with previously reported values for related large area molecular junctions.