Modeling elastic and photoassisted transport in organic molecular wires: Length dependence and current-voltage characteristics

Using a $\pi$-orbital tight-binding model, we study the elastic and photoassisted transport properties of metal-molecule-metal junctions based on oligophenylenes of varying lengths. The effect of monochromatic light is modeled with an ac voltage over the contact. We first show how the low-bias transmission function can be obtained analytically, using methods previously employed for simpler chain models. In particular, the decay coefficient of the off-resonant transmission is extracted by considering both a finite-length chain and infinitely extended polyphenylene. Based on these analytical results, we discuss the length dependence of the linear-response conductance, the thermopower, and the light-induced enhancement of the conductance in the limit of weak intensity and low frequency. In general, the conductance enhancement is calculated numerically as a function of the light frequency. Finally, we compute the current-voltage characteristics at finite dc voltages and show that in the low-voltage regime, the effect of low-frequency light is to induce current steps with a voltage separation determined by twice the frequency. These effects are more pronounced for longer molecules. We study two different profiles for the dc and ac voltages, and it is found that the results are robust with respect to such variations. Although we concentrate here on the specific model of oligophenylenes, the results should be qualitatively similar for many other organic molecules with a large enough electronic gap.

Figure 1

(Color online) A finite block chain of length $N=3$ connected to electrodes at its two ends. This gives rise to self-energies ${\mathbf{\Sigma }}_{11}$ and ${\mathbf{\Sigma }}_{NN}$ on the terminating blocks.

Figure 2

(Color online) A finite chain of length $N=3$ connected to electrodes at its two ends. This gives rise to self-energies ${\mathbf{\Sigma }}_{11}$ and ${\mathbf{\Sigma }}_{NN}$ on the end sites. The nearest-neighbor hoppings inside the ring $(-\gamma )$ and between the rings $(-\eta )$ are different. The lower part indicates also the numbering of the $M=6$ carbon atoms within a ring.

Figure 3

(Color online) Phenyl-ring chains: (a) a periodic chain with $N$ units and (b) an infinite chain. Case (b) is obtained from (a) in the limit $N\to \infty$.

Figure 4

(Color online) (a) The coordinates of the carbon atoms in the direction $z$ along the molecular wire. The left electrode is at $z=0$ and the length of a phenyl-ring unit is $d$. (b) Relative variation of the on-site energies for two different voltage profiles, A and B. The profile function $P\left(z\right)$ describes how the harmonic voltage $V\left(t\right)=V+V_{ac}\cos \left(\omega t\right)$ is assumed to drop over the junction, the voltage at $z$ being given by $V(z,t)=V\left(t\right)P\left(z\right)$.

Figure 5

(Color online) Transmission functions for the oligophenylene wires with lengths $N=1,3,5,7$. The parameters are $\Gamma ∕\gamma =0.5$, $\phi =40°$, and $E_F∕\gamma =-0.4$, as discussed in the text.

Figure 6

(Color online) Dependence of observables on the number of units $N$: (a) conductance, (b) Seebeck coefficient, and (c) the light-induced relative conductance enhancement. The circles correspond to values extracted from the $\tau \left(E\right)$ function (Fig. 5) using Eqs. (44, 45, 46). The red lines correspond to the simple order-of-magnitude estimates of Eqs. (47, 48, 49), with the analytically calculated $\beta \left(E\right)$. In (c), the crosses (× for profile A and + for profile B) show numerical results with the finite values $\alpha =0.5$ and $\hslash \omega ∕\gamma =0.05$ (see Sec. 4C).

Figure 7

(Color online) Zero-bias conductance for different driving frequencies $\omega$ and driving strengths $\alpha =e{V}_{ac}∕\hslash \omega$. Panels (a)–(d) are for $N=1,\dots ,4$. The solid lines correspond to profile A, and the dashed lines to profile B. The lower pair of curves is for $\alpha =0.5$, and the upper pair for $\alpha =2.0$.

Figure 8

Dependence of the conductance on $N$. Circles represent the conductance in the absence of light, while the squares are for light with $\hslash \omega ∕\gamma =0.5$ and $\alpha =1.5$. The solid line is for profile A and the dashed line for profile B.

Figure 9

(Color online) (a) $I\text{−}V$ characteristics $(N=5)$ with and without light for profiles A (solid lines) and B (dashed lines). Results in the presence of light with $\alpha =1.5$ and $\hslash \omega ∕\gamma =0.075$ are indicated with an arrow. (b) The corresponding differential conductances. (c) Same as (b), but concentrating on the low-bias regime and on a logarithmic scale. The vertical dashed lines indicate the approximate positions of the main peak and the light-induced side peaks. They are all separated by $2\hslash \omega ∕e$ in voltage.

Figure 10

(Color online) The voltage-dependent transmission function at three voltages for the wire with $N=5$ and profile B. For profile A, the result is independent of voltage and equal to $\tau (E,V=0)$.

Figure 11

(Color online) Same as Fig. 9c but for wires with $N=1,\dots ,4$.