Length-dependent conductance and thermopower in single-molecule junctions of dithiolated oligophenylene derivatives: A density functional study

We study theoretically the length dependence of both conductance and thermopower in metal-molecule-metal junctions made up of dithiolated oligophenylenes contacted to gold electrodes. We find that while the conductance decays exponentially with increasing molecular length, the thermopower increases linearly as suggested by recent experiments. We also analyze how these transport properties can be tuned with methyl side groups. Our results can be explained by considering the level shifts due to their electron-donating character as well as the tilt-angle dependence of conductance and thermopower. Qualitative features of the substituent effects in our density functional calculations are explained using a tight-binding model. In addition, we observe symmetry-related even-odd transmission channel degeneracies as a function of molecular length.

Figure 1

(Color online) The molecules studied in this work. When they are contacted to the gold electrodes, sulfur atoms replace the terminal hydrogen atoms.

Figure 2

(Color online) HOMO and LUMO energies of the isolated molecules in Fig. 1.

Figure 3

(Color online) (a) The contacts to the gold electrodes are formed through sulfur atoms bonded to the hollow position of the tip of a [111] pyramid. (b) For the transport calculations, the tips of the pyramids are part of the extended molecule (C). The more remote parts (blue) are absorbed into ideal semi-infinite left (L) and right (R) surfaces (gray).

Figure 4

(Color online) [(a) and (b)] Transmission function and the negative of its logarithmic derivative. [(c) and (d)] The corresponding $G$ and $Q$, including the temperature corrections in Eq. (1). The straight lines are best fits to the numerical results for R (solid line), S (dashed line), and D (dashed-dotted line). The experimental data in (d) are from Ref. 14.

Figure 5

(Color online) The parameters of the TB model represented by a molecule with $N=3$ phenyl rings. Also shown are schematic graphs of the on-site energies ${ϵ}_j$ of the different rings $j=1,\dots ,N$ for R, S, and D type molecules with $N=1,\dots ,4$ rings.

Figure 6

(Color online) Influence of methyl substituents on the level alignment of frontier molecular orbitals in benzene. From left to right the molecular structure is displayed together with isosurface plots of the $\text{HOMO}-1$, HOMO, LUMO, and $\text{LUMO}+1$ wave functions. Below these plots the common name of the molecules and the energies of the molecular levels are indicated.

Figure 7

(Color online) (a) Transmission functions for the TB model in Fig. 5 with parameters chosen as explained in the text. (b) Corresponding Seebeck coefficients according to Eq. (2). The straight lines are best fits as in Fig. 4. [(c) and (d)] $G$ and $Q$ as a function of the tilt angle for the molecules with $N=3$. The vertical dotted lines indicate the “equilibrium angles” ${\phi }_{\text{R}}$, ${\phi }_{\text{S}}$, and ${\phi }_{\text{D}}$; the arrows represent changes when going from R3 to S3 to D3.

Figure 8

(Color online) (a) Tilt-angle-dependent transmission $\tau ={\sum }_i{\tau }_i$ resolved in its transmission channels ${\tau }_i$ for the molecule R2 and (b) thermopower for R2, S2, and D2. The curve for R2 is fitted by a function of the form $q_1+q_2{\text{cos}}^2\phi$ with $q_1=13.27\mu \text{V}/\text{K}$ and $q_2=1.38\mu \text{V}/\text{K}$ for the tilt-angle interval from $0°$ to $60°$. The dotted vertical lines indicate DFT equilibrium tilt angles ${\phi }_{\text{R}2}$, ${\phi }_{\text{S}2}$, and ${\phi }_{\text{D}2}$ .

Figure 9

(Color online) Importance ${\tau }_1/{\tau }_2$ of the second transmission channel ${\tau }_2$ as compared to the first ${\tau }_1$ as a function of molecular length for the molecules of series R, S, and D.