Route towards the optimization at given power of thermoelectric heat engines with broken time-reversal symmetry

We investigate the performance at a given power of a thermoelectric heat engine with broken time-reversal symmetry, and derive analytically the efficiency at a given power of a thermoelectric generator within linear irreversible thermodynamics. A universal bound on the efficiency of the thermoelectric heat engine is achieved under a strong constraint on the Onsager coefficients, and some interesting features are further revealed. Our results demonstrate that there exists a trade-off between efficiency and power output, and the efficiency at a given power may surpass the Curzon-Ahlborn limit due to broken time-reversal symmetry. Moreover, optimal efficiency at a given power can be achieved, which indicates that broken time-reversal symmetry offers physically allowed ways to optimize the performance of heat engines. Our study may contribute to the interesting guidelines for optimizing actual engines.

Figure 1

Schematic sketch of a nanoscale thermoelectric device in the presence of a magnetic field B.

Figure 2

$\mathrm{\Lambda }\left(x\right)$ is plotted as a function of the asymmetry parameter $x$. The red dashed line is the vertical asymptote at $x=1$.

Figure 3

Ratio $\eta /{\eta }^{★}$ as a function of $\xi$ for different values of $y$: (a) $y<0$ and (b) $y>0$. The relative power $P/P^{★}$ vs $\xi$ is also shown (green solid line).

Figure 4

Ratio $\eta /{\eta }^{★}$ as a function of $\delta P$ for different values of $y$: (a) $\xi ={\xi }_{+}$ and (b) $\xi ={\xi }_{-}$.

Figure 5

The efficiency ${\eta }^b/{\eta }_c$ as a function of the asymmetry parameter $x$ and the power gain $\delta P$, together with the relevant cross sections for different branches. Here, the favorable case is shown in the left column, and the unfavorable case is shown in the right column. The intersecting points of the pink solid lines and the efficiency (${\eta }^b/{\eta }_c$) curves in (a1) and (b1) signify the CA limit 1/2. The orange lines show the efficiency ${\eta }^b/{\eta }_c$ vs $\delta P$ for $x=1$ in (a2) and (b2).

Figure 6

The efficiency gain $\delta {\eta }^b$ as a function of the power gain $\delta P$ for different values of $x$: (a) $\xi ={\xi }_{+}$ and (b) $\xi ={\xi }_{-}$.