Density functional theory based electron transport study of coherent tunneling through cyclic molecules containing Ru and Os as redox active centers

In our theoretical study in which we combine a nonequilibrium Green's function approach with density functional theory we investigate branched compounds containing Ru or Os metal complexes in two branches, which due to their identical or different metal centers are symmetric or asymmetric. In these compounds the metal atoms are connected to pyridyl anchor groups via acetylenic and phenyl spacer groups in a metaconnection. We find there is no destructive quantum interference (DQI) feature in the transmission function near the Fermi level for the investigated molecules regardless of their symmetry, either in their neutral states or in their charged states. We map the structural characteristics of the range of molecules onto a simplified tight-binding model in order to identify the main differences between the molecules in this study and previously investigated ferrocene compounds in order to clarify the structural sources for DQI, which we found for the latter but not for the former. We also find that local charging on one of the branches changes only the conductance by about one order of magnitude, which we explain in terms of the spatial distributions and charge-induced energy shifts of the relevant molecular orbitals for the branched compounds.

Figure 1

Chemical structure of a cyclic ${\mathrm{Ru}}_2(\mathrm{P}{\left({\mathrm{CH}}_3\right)}_2{)}_8{\left({\mathrm{C}}_2{\mathrm{H}}_4\right)}_4{\left({\mathrm{C}}_6{\mathrm{H}}_4\right)}_4$ bis(pyridyl-diacetylide) molecule.

Figure 2

Junction setups containing the molecule in Fig. 1, where for the double-branched molecule (left panel) the two metal atoms are varied as Ru/Ru, Ru/Os, and Os/Os and for the single-branched molecule (right panel) the single metal atom is either Ru or Os.

Figure 3

Transmission functions from NEGF-DFT calculations for the double-branched molecules Ru/Ru (solid red line), Os/Os (solid black line), and Ru/Os (green solid line) as well as for single-branched Ru (dashed red line) and Os (dashed black line) in the junction setup shown in Fig. 2 in their neutral states.

Figure 4

Transmission functions calculated from NEGF-DFT (red curve) and Larsson's formula for the system Ru/Ru, where for the latter the blue curve denotes (HOMO + ${\mathrm{LUMO})}^2$ and the cyan curve denotes ${\mathrm{HOMO}}^2+{\mathrm{LUMO}}^2$.

Figure 5

Transmission functions for a single-branched Ru molecule, where the red curve is from a NEGF-DFT calculation. The magenta curve is obtained from a NEGF-TB calculation with one FO anchor state on each side and one bridge FO, where the anchor FO is obtained from a subdiagonalization of the transport Hamiltonian containing the $p_z$ state of each carbon or nitrogen atom. For the electrodes a chain of single AOs with on-site energies of 0.83 eV and coupling values of $-5.67$ eV within the electrodes is used.

Figure 6

Spatial distributions of the three FOs in the simplified model for (a) single-branched Ru with the anchor FO on each side at 1.3 eV and the bridge FO at $-0.2$ eV and (b) single-branched Fc with the anchor FO on each side at 1.05 eV and the bridge FO at 1.66 eV [21].

Figure 7

Transmission functions obtained from NEGF-TB calculations using the 3FO model of the single-branched (a) Fc molecule (black solid line) and (b) Ru molecule (black solid line), where the symbols F and R represent Fc and Ru, respectively, and the indices 1–3 refer to the numbers in Table 2. In (a) we start from the parametrization of Fc in Table 2 (${\mathrm{F}}_1{\mathrm{F}}_2{\mathrm{F}}_3$) and replace each of the parameters individually with the one corresponding to Ru and calculate $T\left(E\right)$ for ${\mathrm{R}}_1{\mathrm{F}}_2{\mathrm{F}}_3$ (green line), ${\mathrm{F}}_1{\mathrm{R}}_2{\mathrm{F}}_3$ (blue line), and ${\mathrm{F}}_1{\mathrm{F}}_2{\mathrm{R}}_3$ (magenta line). In (b) we permute the parameters in the opposite direction, starting from the parametrisation of Ru in Table 2 (${\mathrm{R}}_1{\mathrm{R}}_2{\mathrm{R}}_3$) and obtaning $T\left(E\right)$ for ${\mathrm{F}}_1{\mathrm{R}}_2{\mathrm{R}}_3$ (green line), ${\mathrm{R}}_1{\mathrm{F}}_2{\mathrm{R}}_3$ (blue line), and ${\mathrm{R}}_1{\mathrm{R}}_2{\mathrm{F}}_3$ (magenta line).

Figure 8

$E_0$ versus (a) the energy difference $\mathrm{\Delta }\varepsilon$ between the anchor and bridge states and (b) the coupling value $t_{L/R}$ and (c) $t_D$ for Ru (black curve) and Fc (red curve). In (a) the $E_0$ values resulting from $\mathrm{\Delta }\varepsilon$ (Ru) and $\mathrm{\Delta }\varepsilon$ (Fc) at $-1.5$ and 0.6 eV are marked as black and red dots, respectively, while the values for $t_{L/R}$ and $t_D$ are defined by the color of the line the dots are situated on.

Figure 9

Transmission functions calculated with NEGF-TB for the single-branched Ru system for (a) $t_D=0.00005$ eV (black line), 0.0004 eV (red line), 0.002 eV (green line), and 0.09 eV (blue curve) and (b) $t_D=-0.00005$ eV (black line), $-0.0001$ eV (red line), $-0.008$ eV (green line), and $-0.2$ eV (blue line).

Figure 10

Junction geometry for two neighboring cells in the periodic setup for the scattering region of double-branched Ru/Os containing a chlorine atom for achieving local charging with a distance of 5.2 Å between Cl and the closer-lying metal atom.

Figure 11

(a) Electron occupation of the Ru $d$ orbitals in a low-spin configuration for neutral and charged molecules according to ligand field theory, (b) spatial localization of the four relevant occupied orbitals, and (c) their corresponding energies within the junction and with respect to the Fermi level for three systems, namely, Ru/Ru, Ru/Os, and Os/Os in their neutral and charged states.

Figure 12

Transmission functions for (a) Ru/Ru, (b) Ru/Os, and (c) Os/Os in their respective neutral (black solid line) and charged states (red solid line), where for the latter the average of spin-up and spin-down contributions has been used.