Quantum control of nonlinear thermoelectricity at the nanoscale

We theoretically study how one can control and enhance nonlinear thermoelectricity by regulating quantum coherence in nanostructures such as a quantum dot system or a single-molecule junction. In nanostructures, the typical temperature scale is much smaller than the resonance width, which largely suppresses thermoelectric effects. Yet we demonstrate one can achieve a reasonably good thermoelectric performance by regulating quantum coherence. Engaging a quantum-dot interferometer (a quantum dot embedded in the ring geometry) as a heat engine, we explore the idea of thermoelectric enhancement induced by the Fano resonance. We develop an analytical treatment of fully nonlinear responses for a dot with or without strong interaction. Based on the microscopic model with the nonequilibrium Green function technique, we show how to enhance efficiency and/or output power as well as where to locate an optimal gate voltage. We also argue how to assess nonlinear thermoelectricity by linear-response quantities.

Figure 1

(a) A schematic of a quantum-dot interferometer. The hopping along the direct conducting channel between the two reservoirs (the gray region) can be adjusted. (b) The setting of electrical and thermal voltages for a heat engine: $T_L>T_R$ and ${\mu }_L<{\mu }_R$.

Figure 2

The figure of merit $ZT$ at $k_{B}T=0.2\gamma$, as a function of $\mu -{\epsilon }_d$ by changing (a) $x=0$ ($q=\infty$), (b) $x=0.01$ ($q=4.95$), (c) $x=0.1$ ($q=1.42$) and (d) $x=0.3$ ($q=0.64$). Red and blue indicate corresponding thermopower is positive or negative. Green lines show $ZT$ at the zero-temperature limit. Dotted lines correspond to the normalized conductance.

Figure 3

Density plots of the figure of merit $ZT$ and the linear-response estimate of the maximal power ${\mathcal{P}}_{max}$, as a function of $\mu -{\epsilon }_d$ and $x$. (a) $ZT$ at $k_{B}T=0.2\gamma$, (b) $ZT$ at $k_{B}T=\gamma$, (c) ${\mathcal{P}}_{max}$ at $k_{B}T=0.2\gamma$, (d) ${\mathcal{P}}_{max}$ at $k_{B}T=\gamma$. Green dashed lines specify the location of the Fano node.

Figure 4

Linear-response estimate of the power-efficiency diagram at (a) $k_{B}T=0.2\gamma$ and (b) $k_{B}T=\gamma$, by changing $x$: $x=0.0$ (red), $x=0.01$ (green), $x=0.1$ (blue), and $x=0.3$ (orange).

Figure 5

Evolution of nonlinear efficiencies $\eta /{\eta }_C$ and the output powers $\mathcal{P}$ by changing gate and bias voltages. The setting is $k_B{T}_L=2k_B{T}_R=0.2\gamma$ (with ${\eta }_C=0.5$). We normalize $\mathcal{P}$ by ${\mathcal{P}}_{\mathrm{\Delta }T}$. (a) $\eta /{\eta }_C$ at $x=0.01$; (b) $\eta /{\eta }_C$ at $x=0.1$; (c) $\mathcal{P}$ at $x=0.01$; (d) $\mathcal{P}$ at $x=0.1$.

Figure 6

Power-efficiency diagrams of Fig. 5. Color lines correspond to the evolution by changing bias voltage with a fixed gate voltage, and dashed lines correspond to the one by changing the gate voltage with a fixed bias voltage. (a) $x=0.01$; (b) $x=0.1$.

Figure 7

Comparison of the power-efficiency diagrams at different $k_B{T}_R$. Other parameters are the same with Figs. 5 and 6. The power output is normalized by ${\mathcal{P}}_{\mathrm{\Delta }T}$ for both cases. (a) $T_R=0.5T_L({\eta }_C=0.50)$; (b) $T_R=0.05T_L({\eta }_C=0.95)$.

Figure 8

Effective transmission $\mathcal{T}\left(\mu \right)$ for an interacting dot as a function of $\mu -{\epsilon }_d$ by changing $U$: $U/\gamma =0$ (dashed gray lines), 2 (red lines), and 4 (blue line). The results are shown for (a) $x=0$ and (b) $x=0.1$.

Figure 9

The evolution of the efficiency $\eta /{\eta }_C$ and the power output $\mathcal{P}$ at $x=0.1$ by changing $U$ at $k_B{T}_L=2k_B{T}_R=0.2\gamma$ with ${\eta }_C=0.5$. The power $\mathcal{P}$ is normalized by ${\mathcal{P}}_{\mathrm{\Delta }T}$. (a) $\eta /{\eta }_C$ at $U=2\gamma$; (b) $\eta /{\eta }_C$ at $U=4\gamma$; (c) $\mathcal{P}$ at $U=2\gamma$; (d) $\mathcal{P}$ at $U=2\gamma$.

Figure 10

Power-efficiency diagram by changing $U/\gamma =2$, 4, 8. Other parameters are the same as in Fig. 9. Shaded regions correspond to the evolution for $x=0$ (red), $x=0.01$ (green), $x=0.1$ (blue), and $x=0.3$ (orange). (a) $U=2\gamma$; (b) $U=4\gamma$; (c) $U=8\gamma$.

Figure 11

Nonlinear flows of the particle and the heat at $x=0.1$ as a function of bias voltage. Parameters are the same as in Fig. 5. Different colored lines correspond to different values of the gate voltage from ${\mu }_L-{\varepsilon }_d=-2.0\gamma$ to $+0.5\gamma$. (a) Particle flow $I_L$; (b) Heat flow $J_L^Q$.

Figure 12

Comparison of the linear thermopower at $T_{\text{op}}$ with (a) nonlinear efficiency of Fig. 5 and (b) the case where the dot-reservoir couplings are highly asymmetric. (a) ${\gamma }_L={\gamma }_R=0.5\gamma$; (b) ${\gamma }_L=0.05\gamma ,{\gamma }_R=0.95\gamma$.

Figure 13

Linear-response estimate of the power-efficiency diagram using $T_{\text{op}}$, for (a) ${\eta }_C=0.50$ and (b) ${\eta }_C=0.95$. Parameters are chosen to correspond with Fig. 7. (a) ${\eta }_C=0.50$; (b) ${\eta }_C=0.95$.