Efficiency of three-terminal thermoelectric transport under broken time-reversal symmetry

We investigate thermoelectric efficiency of systems with broken time-reversal symmetry under a three-terminal transport. Using a model of Aharonov-Bohm interferometer formed with three noninteracting quantum dots, we show that Carnot efficiency ${\eta }_C$ can be achieved when the thermopower is a symmetric function of the applied magnetic field. On the other hand, the maximal value of the efficiency at maximum power is obtained for asymmetric thermopower. Indeed, we show that the Curzon-Ahlborn limit is exceeded within the linear response regime in our model. Moreover, we investigate thermoelectric efficiency for random Hamiltonians drawn from the Gaussian unitary ensemble and for a more abstract transmission model. In this latter model, we find that the efficiency is improved using sharp energy-dependent transmission functions.

Figure 1

Schematic picture of Aharonov-Bohm interferometer model.

Figure 2

Optimized values of maximum efficiency ${\eta }_{\max }$ as a function of the asymmetry parameter $x$ for the Aharonov-Bohm interferometer discussed in the text, with a linear [panel (a)] and a logarithmic [panel (b)] scale for ${\eta }_{\max }$. Note that ${\eta }_{\max }$ is scaled in units of ${\eta }_C$ in this figure and thereafter.

Figure 3

Optimized values of efficiency at maximum power $\eta \left({\omega }_{\max }\right)$ versus asymmetry parameter $x$ for the Aharonov-Bohm interferometer. The inset shows the variation of $\eta \left({\omega }_{\max }\right)$ near the symmetric value $x=1$. Note that the Curzon-Ahlborn limit $\eta \left({\omega }_{\max }\right)={\eta }_C/2$ is exceeded in this regime. $\eta \left({\omega }_{\max }\right)$ is scaled in units of ${\eta }_C$ in this figure and thereafter.

Figure 4

Relation between maximum efficiency ${\eta }_{\max }$ and asymmetry parameter $x$ [panel (a)] and between efficiency at maximum power $\eta \left({\omega }_{\max }\right)$ and asymmetry parameter [panel (b)] for $20000$ realizations of $3\times 3$ Hamiltonian drawn from the Gaussian unitary ensemble.

Figure 5

Distribution of (a) maximum efficiency ${\eta }_{\max }$, (b) efficiency at maximum power $\eta \left({\omega }_{\max }\right)$, and (c) asymmetry parameter $x$. Inset of (a) and (b) shows the distribution of ${\eta }_{\max }$ and $\eta \left({\omega }_{\max }\right)$, taking into account only data such that $|x-1|>0.5$. Note that obtaining high values of both efficiency and asymmetry is highly improbable.

Figure 6

Maximum efficiency ${\eta }_{\max }$ versus asymmetry parameter $x$ under three-terminal transport with transmission model described in the text. Solid curve corresponds to the theoretically predicted upper bound ${\eta }_{\max }^{☆}$ [given by Eq. (15)] for ${\eta }_{\max }$ for a generic broken time-reversal symmetry system. The optimized efficiency saturates the theoretical upper bound only near the symmetric value $x=1$.

Figure 7

Optimized transmission functions for $x=1.4$ using the transmission model. Panel (a) shows $-f^{\prime }$, with $f^{\prime }$ derivative of the Fermi distribution function $[f^{\prime }\left(E\right)=\frac{\partial f\left(E\right)}{\partial E}]$, as a function of energy $E$ with optimal temperature $T=1$ and chemical potential $\mu =1.3333$. The transmission probabilities ${\tau }_{LR}$, ${\tau }_{RL}$, ${\tau }_{LP}$, ${\tau }_{PL}$, ${\tau }_{RP}$, and ${\tau }_{PR}$ at each energy $E$ are plotted in panels (b), (c), (d), (e), (f), and (g), respectively. Note that optimal transmission probabilities are either 0 or 1.

Figure 8

Efficiency at maximum power $\eta \left({\omega }_{\max }\right)$ versus asymmetry parameter $x$ under three-terminal transport using the transmission model described in the text. Theoretically predicted upper bound $\eta {\left({\omega }_{\max }\right)}^{☆}$ [given by Eq. (16)] of $\eta \left({\omega }_{\max }\right)$ is plotted as solid curve. Dotted dashed line corresponds to the Curzon-Ahlborn limit $\eta \left({\omega }_{\max }\right)={\eta }_C/2$. Note that the Curzon-Ahlborn limit is exceeded in the interval [1, 2] with the transmission model.