Bound on thermoelectric power in a magnetic field within linear response

For thermoelectric power generation in a multiterminal geometry, strong numerical evidence for a universal bound as a function of the magnetic-field induced asymmetry of the nondiagonal Onsager coefficients is presented. This bound implies, inter alia, that the power vanishes at least linearly when the maximal efficiency is approached. In particular, this result rules out that Carnot efficiency can be reached at finite power, which an analysis based on the second law only would, in principle, allow.

Figure 1

Sketch of a thermoelectric generator. Both heat and particle current run from the hot reservoir on the left to the cold reservoir on the right, thereby traversing the device D, which is represented by the dashed box. Within the multiterminal setup, the device consists of a central scattering region subject to a magnetic field $\mathbf{B}$ and $n-2$ probe terminals. The figure corresponds to $n=4$.

Figure 2

Scatter plots of the rescaled Onsager coefficient ${\overline{L}}_{qq}$ as a function of the ratio $y/h_n\left(x\right)$ for respectively $50000$ randomly chosen $n$-terminal models. The dashed line is given by ${\overline{L}}_{qq}=1-y/h_n\left(x\right)$. To generate these data, we used the set of permutations $\left\{\right(1\left)\right(2,3),(1,3\left)\right(2),(1,3,2\left)\right\}$ for $n=3$ (left column) and the set $\left\{\right(1\left)\right(2\left)\right(3,4),(1,4\left)\right(2,3),(1,4,2,3\left)\right\}$ for $n=4$ (right column). While for all plots the sign of the $w_k$ is chosen randomly, $|w_k|$ is sampled from an increasingly large interval $\Delta \subseteq [0,1]$. Specifically, we have from the first to the fourth line $\Delta =[1,1]$, $\Delta =[0.99,1]$, $\Delta =[0.5,1]$, $\Delta =[0,1]$.

Figure 3

Scatter plots of the rescaled Onsager coefficient ${\overline{L}}_{qq}$ as a function of the ratio $y/h_n\left(x\right)$ for respectively $50000$ randomly chosen $n$-terminal models. For all plots, the sign of the $w_k$ is chosen by random and their absolute value has been set to $|w_k|=1$. The sets of permutations used to generate the shown data are $\left\{\right(1,3\left)\right(2\left)\right(4,5),(1\left)\right(2,4\left)\right(3,5),(1,4,3,2,5\left)\right\}$ for $n=5$, $\left\{\right(1\left)\right(2\left)\right(3,6\left)\right(4,5),(1,6,4,2,5,3),(1,6,4\left)\right(2,3,5\left)\right\}$ for $n=6$, $\left\{\right(1,3\left)\right(2,4\left)\right(5\left)\right(6,7),(1,5,7\left)\right(2,6\left)\right(3,4),(1,7,4,5,2,6,3\left)\right\}$ for $n=7$, and $\left\{\right(1,6\left)\right(2,8\left)\right(3,4,7,5),(1,7,6,3,2,8,5\left)\right(4),(1,5,4)$ $(2,3)(6,8)\left(7\right)\}$ for $n=8$.

Figure 4

Bound (28) on the rescaled maximum power output ${\widehat{P}}_{\max }\left(x\right)/P_0$ of the multiterminal thermoelectric generator as a function of the asymmetry parameter $x$ for $n=3$, $n=4$, and $n\to \infty$.

Figure 5

Bounds on the power ${\widehat{P}}_{\max }(\eta ,x)$ of the $n$-terminal model as a thermoelectric heat engine in units of $P_0$ and as functions of the asymmetry parameter $x$ and the efficiency $\eta$ for $n=3,4$ and $n\to \infty$. The white regions in the plots are forbidden by the bound (15).

Figure 6

Plots of the bound (33) on the power at maximum efficiency of a $n$-terminal thermoelectric generator in units of $P_0$ and as functions of the asymmetry parameter $x$ and the maximum efficiency ${\eta }_{\max }$ for $n=3$, $n=4$, and $n\to \infty$.