Microscopic theory, analysis, and interpretation of conductance histograms in molecular junctions

Abstract

Molecular electronics break-junction experiments are widely used to investigate fundamental physics and chemistry at the nanoscale. Reproducibility in these experiments relies on measuring conductance on thousands of freshly formed molecular junctions, yielding a broad histogram of conductance events. Experiments typically focus on the most probable conductance, while the information content of the conductance histogram has remained unclear. Here we develop a microscopic theory for the conductance histogram by merging the theory of force-spectroscopy with molecular conductance. The procedure yields analytical equations that accurately fit the conductance histogram of a wide range of molecular junctions and augments the information content that can be extracted from them. Our formulation captures contributions to the conductance dispersion due to conductance changes during the mechanical elongation inherent to the experiments. In turn, the histogram shape is determined by the non-equilibrium stochastic features of junction rupture and formation. The microscopic parameters in the theory capture the junction's electromechanical properties and can be isolated from separate conductance and rupture force (or junction-lifetime) measurements. The predicted behavior can be used to test the range of validity of the theory, understand the conductance histograms, design molecular junction experiments with enhanced resolution and molecular devices with more reproducible conductance properties.

Fig. 1. Break-junction experiments.

In these experiments a metallic tip is brought into contact with a metallic surface and the overall length of the junction L is increased by applying an external force F until rupture at an elongation L_f. a The rupture of the metal-metal junction leads to an initial electrode–electrode gap ξ₀ in which the molecule is anchored forming a molecular junction. b The pulling of the molecular junction results in its rupture at electrode gap ξ_r. Both ξ₀ and ξ_r are stochastic variables determined by rupture statistics. The circled numbers in a, b signal the steps into which the process has been divided, as described in Section “Results”. c The metal–metal rupture can be seen as the rupture of two brittle springs connected in series. The blue and red springs represent the electrodes (surface and cantilever in a scanning tunneling microscope break-junction experiment). Increasing the overall junction length L from contact to rupture by ΔL_f leaves an identical junction gap ξ₀. d The molecular junction rupture can be represented by the rupture of three brittle springs connected in series, where the purple spring represents the molecule.

Fig. 2. Schematic representation of the free-energy profile (FEP) of a metal-metal or molecular junction along the pulling coordinate.

Here ξ is the electrode gap and ξ_eq its value at mechanical equilibrium. External mechanical forces (F) decrease (red line, F > 0) or increase (blue line, F < 0) the free-energy barrier ΔA^‡ (which is located a distance χ^‡ away from ξ_eq) between the unruptured and ruptured states with respect to that of the pristine junction ($\mathrm{\Delta }{A}_0^{‡}$).

Fig. 3. Modeling of conductance histogram in a break-junction experiment.

a Probability density function of the initial (p₀(ξ₀), Eq. (3)) and rupture (p_r(ξ_r), Eq. (4)) electrode gaps ξ in a molecular junction. b Probability of visiting the electrode gap ξ (P(ξ)) during a break-junction experiment. The dotted lines represent the probability that the junction has been formed (P₀(ξ))/has not been ruptured (1 − P_r(ξ)) at a given ξ. c Conductance G histogram calculated with Eq. (8). The histogram is reported on a log scale and in units of the quantum of conductance G₀ = 2e²/h. In all cases, the parameters in Table 1 were used.

Fig. 4. Influence of the microscopic parameters and loading rate on the conductance G histogram.

a Effect of the base transmission T₀, b transmission decay coefficient γ, and c loading rate $\stackrel{°}{F}$ on the break-junction conductance histograms, calculated with Eq. (8). For these cases, all parameters except the one being varied are those in Table 1. The histograms are reported on a log scale and in units of the quantum of conductance G₀. The influence of the remaining microscopic parameters is included in Fig. S2.

Fig. 5. Utility of Eq. (8) in fitting experimental conductance G histograms.

The plot shows experimental conductance histograms of junctions formed with a–c aliphatic molecules (Cn-SMe), d–f aromatic molecules (An-(N or SMe)), g–i metallofullerene complexes (Sc2C2@Cn), j–l radical containing molecules of varying length and charge (Bn^m+) and their accurate fit to Eq. (8). The values of the fitting parameters are shown in Table 2. The histograms are reported on a log scale and in units of the quantum of conductance G₀. Experimental data was provided by Prof. Venkataraman for Cn-SMe and obtained from refs. ^– for the other cases.

Fig. 6. Bimodal experimental conductance G histograms corresponding to the Au-Cn-DT-Au junctions and their fit to the p1(logT)+p2(logT) equation.

Here, both $p_1\left(\log T\right)$ (in blue) and $p_2\left(\log T\right)$ (in red) correspond to Eq. (8) with different fitting parameters shown in Table 2. The fitting of bimodal distributions allows us to identify individual high/low transmission peaks. The histograms are reported on a log scale and in units of the quantum of conductance G₀. The experimental data was provided by Professor Latha Venkataraman.