Nonlinear thermoelectric properties of molecular junctions with vibrational coupling

We present a detailed study of the nonlinear thermoelectric properties of a molecular junction, represented by a dissipative Anderson-Holstein model. A single-orbital level with strong Coulomb interaction is coupled to a localized vibrational mode and we account for both electron and phonon exchange with both electrodes, investigating how these contribute to the heat and charge transports. We calculate the efficiency and power output of the device operated as a heat to electric power converter in the regime of weak tunnel coupling and phonon exchange rate and identify the optimal operating conditions, which are found to be qualitatively changed by the presence of the vibrational mode. Based on this study of a generic model system, we discuss the desirable properties of molecular junctions for thermoelectric applications.

Figure 1

(Color online) (a) Sketch of a thermoelectric junction with a single molecule, drawn by way of illustration as ${\text{C}}_{60}$. Operated as a thermal to electric power converter, a temperature bias is applied across the device. For an energy level above the electrochemical potentials, $ϵ>{\mu }_r$, this can drive a net flow of electrons from the hot (h) to the cold (c) electrode by tunneling through the junctions. In addition, combined electron and phonon heat currents $Q_h$ and $Q_c$ are driven through the molecule. (b) The thermoelectric circuit is loaded by a resistor $R$, and as a result a voltage bias is applied to the cold electrode, partially opposing the thermally induced electron flow. The voltage thus ranges from 0 (corresponding to $R=0$) to the value where $I=0$ (corresponding to $R\to \infty$). In the rest of the paper we will however consider a test device as drawn in (a), where $V$, rather than $R$, is the free parameter. We investigate how to adjust $V$ and the other parameters to obtain a maximal efficiency $\eta$ and output power $P$.

Figure 2

(Color online) [(a)–(c)] Efficiency $\eta$ at thermal bias $\Delta T=T$, as function of voltage bias $V$ and level position $ϵ$ for increasing coupling to substrate phonons, (a) $\gamma =0$, (b) $\gamma =\Gamma /10$, and (c) $\gamma =\Gamma$. In all plots $\lambda =1$, $\omega =T$, and $U=10T$, and the couplings are symmetric, ${\gamma }_h={\gamma }_c=\gamma$ and ${\Gamma }_h={\Gamma }_c=\Gamma$. (d) Output power $P$ as function of $V$ and $ϵ$ for the parameters used in (b) (the power depends only weakly on $\gamma$).

Figure 3

(Color online) (a) Efficiency $\eta$, (b) output power $P$, and (c) molecular temperature $T_M$, as function of $V$ and $\Delta T$, with $ϵ=5T$ and other parameters as in Fig. 2b. Along the edge of the white areas in (a) and (b) $I=0$, and this edge therefore defines the nonlinear thermopower $S\left(\Delta T\right)=-V/\Delta T$ at $I=0$. The standard linear response thermopower, $S=S\left(\Delta T\to 0\right)$, is thus given by the slope close to zero, indicated by the green dashed line in (a). Note that $T_M$ in (c) is plotted also in the regime where the current has been reversed, even though the device would normally not be operated under such conditions. (d) $\eta$ (thick lines) and $T_M$ (thin lines, normalized by $10T$) as function of $\Delta T$ at $V=2.5T$ for ${\gamma }_h={\gamma }_c=\Gamma /10$ (red solid lines), $10{\gamma }_h={\gamma }_c=\Gamma$ (green dashed lines), and ${\gamma }_h={\gamma }_c=\Gamma$ (blue dotted lines).

Figure 4

(Color online) (a) Maximum efficiency, ${\eta }_{\text{max}}$, as function of $\Delta T$, i.e., the maximum obtained from vertical cuts in a plot such as Fig. 3a. Parameters as in Figs. 3a, 3b, 3c but the strength of the coupling to substrate phonons is varied. Here $\gamma ={\gamma }_h={\gamma }_c$ and $\Gamma ={\Gamma }_h={\Gamma }_c$. (b) Same as (a) but instead showing efficiency at maximum output power $P_{\text{max}}$, i.e., $\eta$ taken at the maximum $P$ obtained from vertical cuts in a plot such as Fig. 3b.

Figure 5

(Color online) Dependence on the molecular vibration frequency $\omega$ and its coupling to the electron charge $\lambda$, all other parameters fixed to those of Figs. 3a, 3b, 3c. [(a) and (b)] (a) Efficiency $\eta$ and (b) power $P$, as function of $V$ and $\Delta T$ for $\lambda =2$ and $\omega =T$. [(c) and (d)] Efficiency $\eta$ as function of $V$ and $\Delta T$ for (c) $\lambda =0.5$ and $\omega =T$ and (d) $\lambda =1$ and $\omega =T/10$.