Strong Bounds on Onsager Coefficients and Efficiency for Three-Terminal Thermoelectric Transport in a Magnetic Field

For thermoelectric transport in the presence of a magnetic field that breaks time-reversal symmetry, a strong bound on the Onsager coefficients is derived within a general setup using three terminals. Asymmetric Onsager coefficients lead to a maximum efficiency substantially smaller than the Carnot efficiency reaching only ${\eta }_C/4$ in the limit of strong asymmetry. Related bounds are derived for efficiency at maximum power, which can become larger than the Curzon-Ahlborn value ${\eta }_C/2$, and for a cooling device. Our approach reveals that in the presence of reversible currents the standard analysis based on the positivity of entropy production is incomplete without considering the role of current conservation explicitly.

Figure 1

Sketch of a thermoelectric device in the presence of a magnetic field $\mathbf{B}$. The entire dashed box represents the conductor $C$, which is connected to two reservoirs. For the special case of the three terminal model, the conductor consists essentially of a scattering region and an additional probe terminal.

Figure 2

Bounds on benchmarks in units of ${\eta }_C$ as functions of the asymmetry parameter $x$. The upper diagram shows ${\eta }_{\max }^{*}$, the lower one $\eta (P_{\max }{)}^{*}$. The solid curves represent the bounds following from relation (15). In the limit $x\to \pm \infty$ both functions asymptotically approach the value $1/4$, which is shown by the dash-dotted lines. For comparison, the bounds obtained by Benenti et al. [12] solely from the second law (3) have been included as dashed curves. The dotted line in the lower panel indicates the Curzon-Ahlborn limit $1/2$.

Figure 3

Bounds on the coefficient of performance of a refrigerator $\epsilon$ in units of ${\eta }_C^r$ as a function of the asymmetry parameter $x$. The dashed line follows from bare second law, the solid line from the stronger relation (15).