Influence of structural disorder and large-scale geometric fluctuations on the coherent transport of metallic junctions and molecular wires

Structural disorder is present in almost all experimental measurements of electronic transport through single molecules or molecular wires. To assess its influence on the conductance is computationally demanding because a large number of conformations must be considered. Here we analyze an approximate recursive-layer Green’s function approach for the ballistic transport through quasi-one-dimensional nanojunctions. We find a rapid convergence of the method with its control parameter, the layer thickness, and good agreement with existing experimental and theoretical data. Because the computational effort rises only linearly with system size, this method permits treatment of very large systems. We investigate the conductance of gold and silver wires of different sizes and conformations. For weak electrode disorder and imperfect coupling between electrode and wire we find conductance variations of approximately 20%. Overall we find the conductance of silver junctions well described by the immediate vicinity of narrowest point in the junction, a result that may explain the observation of well-conserved conductance plateaus in recent experiments on silver junctions. In an application to flexible oligophene wires, we find that strongly distorted conformations that are sterically forbidden at zero temperature, contribute significantly to the observed average zero-bias conductance of the molecular wire.

Figure 1

(Color online) Schematic representation of the different regions of a single molecule junction in the Landauer approach. (a) Definition of the extended molecule including a fraction of the electrodes. This permits the natural charge transfer $\delta n$ and screening effects as an organic system is attached to a metal surface. (b) Realization of a molecular junction by a phenyl-ring-based wire coupled via sulfur atoms to the [111] layers of gold leads. The semi-infinite leads are represented by their surface Green’s function defined on the atoms of the dark shaded area. Horizontal lines indicate a possible division into a left and a right part of the system. (c) Representative division of the system into principal layers to illustrate the recursive Green’s function approach.

Figure 2

(Color online) Convergence test of the conductance depending on the “principal layer” thickness for several nanojunctions. (a) Model nanojunction of $45.2\text{Å}$ length and minimal cross section of one atom allowing for several different principal layer divisions, indicated by the marked lines below the conformation. On the right-hand side the thickness of the principal layers of the actual division is indicated, respectively. $w$ is given in units of the [111] atomic layer distance $d_{\left[111\right]}=2.35\text{Å}$ (b) Corresponding conductance values for the upper described sets of “principal layers” for a silver and a gold contact, respectively. The dependence of the conductance on the layer division is also shown for similar metallic junctions with a minimal cross section of 2, 3, and 4 atoms, respectively. (c) Metal quantum wires with one conductance quantum, but with increasing length between 20 and 32 atomic layers show the same rapid convergence behavior with increasing principal layer thickness.

Figure 3

(Color online) The total transmission $\tau \left(E\right)$ of the model silver junction shown in Fig. 1a, in the different levels of approximation, i.e., a changing number of “principal layers.” Convergence with respect to the principal layer thickness is achieved as soon as $w$ becomes is larger than 2.

Figure 4

(Color online) The tilt- and twinning-angle dependences of the conductance of a silver point contact. (a) Only a moderate change of $⟨G⟩$ is observed during tilting the electrodes up to $70°$. (b) The twinning of the electrodes between $0°$ and $60°$ results in a nearly constant conductance.

Figure 5

Representative examples of the generated conformations of silver nanojunction with an increasing number of surface vacancies. The presence of defects leads to a decrease of the conductance by up to 30%, indicated by the corresponding conductance values below the geometries. The number of vacancies in conformation is given in brackets.

Figure 6

(Color online) Conductance values averaged over conformations with equal number of surface impurities. The error bars indicates that the change in the conductance depends less on the number of defects, but more on their positions.

Figure 7

(Color online) Transmission of silver nanoclusters. (a) Shows the top view on the examined nanocluster conformations with 5, 7, 180, 220, and 260 atoms. (b) Cluster conformations energetically optimized on a silver substrate layer with a pyramidal electrode on top. (c) Calculated transmission function of the junction conformations shown in “b.” The vertical line indicates the Fermi energy.

Figure 8

(Color online) Configurations of the organic molecular wires studied in this work. (a) Oligophenylene molecules covalently bond to ${\text{Au}}_{19}$ clusters along the crystallographic [111] axis. One phenylene ring unit represents one principal layer. (b) Total transmission as a function of the energy of the shown oligophenylene with (dashed line) and without (solid line) the principal layer approximation in good qualitative agreement with the DFT results of Ref. 23. In the layer approximation one principal layer contains a single phenylene ring unit. The vertical line indicates the Fermi energy. (Inset) Length dependence of the conductance of the oligophenylene wires. The conductance decreases exponentially with the number of the phenylene rings in the wire in good agreement with experimental data.

Figure 9

(Color online) Thermal influence on the conductance of a molecular wire at 300 K. (a) Fluctuation of the torsion angles ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$ between the ring units occurring in Au-h-R4, respectively. (b) Distribution histogram of the frequency of occurrence of a particular torsion angle. (c) Corresponding conductance (black) and average torsion angle $\overline{\varphi }$ (gray/turquoise) at the fluctuation process during 10 ps simulation time. (d) Zoom into the 4–5ps range, which shows that a short-time increase of the torsion angles (thin gray/colored curves, left axis) leads to a strong decay of the total transmission (bold black curve, right axis) of the nanowire.