Conductance of molecular wires and transport calculations based on density-functional theory

The experimental value for the zero bias conductance of organic molecules coupled by thiol-groups to gold electrodes tends to be much smaller than the theoretical result based on density functional theory (DFT) calculations, often by orders of magnitude. To address this puzzle we have analyzed the regime within which the approximations made in these calculations are valid. Our results suggest that a standard step in DFT based transport calculations, namely approximating the exchange-correlation potential in quasistatic nonequilibrium by its standard equilibrium expression, is not justified at weak coupling. We propose, that the breakdown of this approximation is the most important source for overestimating the width of the experimentally observed conductance peak and therefore also of the zero bias conductance. We present a numerical study on the conductance of an organic molecule that has recently been studied in experiments that fully agrees with this conclusion.

Figure 1

Schematic representation for a conductance measurement of the molecule $\left(9,10\text{−}\mathrm{Bis}\left(\left(2’\text{−}\mathrm{para}\text{−}\mathrm{mercaptophenyl}\right)\text{−}\mathrm{ethinyl}\right)\text{−}\mathrm{anthracene}\right)$: between gold contacts (Ref. 1).

Figure 2

Transmission of a four atom gold chain. Upper panel, contact was made of 54 gold atoms, see contacts in Fig. 1. Out of these 41 atoms ($◻$, $5\times 5$ and $4\times 4$ bottom layers) have been coupled to external leads. For comparison, results with 29 coupling atoms are shown as well $(\circ )$. Lower panel, same as upper panel with 84 gold atoms defining the contacts. $\left(E_F\approx -5.1\mathrm{eV}\right)$ Traces are (nearly) independent of the modeling parameters chosen, and they agree well with experiments and previous calculations (Ref. 29).

Figure 3

Transmission of the molecule in Fig. 1 with sulfur atom coupling to one (solid), two (dashed), and three (dotted) gold atoms. $\left(E_F\approx -5.1\mathrm{eV}\right)$ Modifications in the atomic contact structure lead to a slight shift of the conductance peak that offer an explanation for the inhomogeneous peak broadening observed in experiments, see Fig. 4.

Figure 4

Upper curves, $dI∕dV$ curve from data in Fig. 3 based on $I\left(V\right)=\int dET\left(E\right)\left(f\left(E+V∕2\right)-f\left(E-V∕2\right)\right)$ with $f\left(E\right)$ denoting the Fermi function $\left(T=300\mathrm{K}\right)$. Lower curves, experimental data from Ref. 1. Different traces represent results from consecutive voltage sweeps. The theoretical and experimetal traces exhibit the same qualitative features, but strong quantitative deviations in the conductance magnitude exist.

Figure 5

Schematic representation of geometric degrees of freedom defining the sulfur-gold coupling (center atom, sulfur; left, gold surface; right, benzene molecule).

Figure 6

Transmission of $\mathrm{benzene}\text{−}1,4\text{−}\mathrm{dithiol}$ for various parameters $\alpha$, $d$, and $\beta$ as defined in Fig. 4. Plot shows, that the transmission is robust against small structural modifications induced, e.g., by strain. Upper panel, molecule, coupling to an $\mathrm{Au}\left(111\right)$ hollow site, fully geometry optimized (solid) and same molecule after rotation (dotted–dashed, $\alpha =\pi ∕24$). Middle, change in bondlength (optimal value, $d=2.55\mathrm{\AA }$), applied change $0.05\mathrm{\AA }$ (dashed) and $0.1\mathrm{\AA }$ (long dashed). Bottom, change in angle $\beta$ by $\pi ∕12$ $\left(E_F\approx -5.1\mathrm{eV}\right)$.

Figure 7

Transmission of $\mathrm{benzene}\text{−}1,4\text{−}\mathrm{dithiol}$ calculated within DFT (solid $\mathrm{Au}54$, as in Fig. 6; dotted-dashed $\mathrm{Au}14$), HF (dashed $\mathrm{Au}14$), and extended Hückel (dotted $\mathrm{Au}14$) with 14 and 54 atom gold clusters simulating the electrodes. DFT and HF calculations show large differences which indicates a breakdown of the EXCA explicated in Sec. 2B.