Efficiency bounds on thermoelectric transport in magnetic fields: The role of inelastic processes

We examine the efficiency of an effective two-terminal thermoelectric device under broken time-reversal symmetry. The setup is derived from a three-terminal thermoelectric device comprising a thermal terminal and two electronic contacts, under a magnetic field. We find that breaking time-reversal symmetry in the presence of the inelastic electron-phonon processes can significantly enhance the figure of merit for delivering electric power by supplying heat from a phonon bath, beyond the one for producing the electric power by investing thermal power from the electronic heat current. The efficiency of such a device is bounded by the non-negativity of the entropy production of the original three-terminal junction. The efficiency at maximal power can be quite close to the Carnot efficiency, but then the electric power vanishes.

Figure 1

Sketch (a) and simplified model (b) of a thermoelectric device, comprising two electronic terminals (held at chemical potentials ${\mu }_L^{}$ and ${\mu }_R^{}$, and at temperatures $T_L^{}$ and $T_R^{}$), and a thermal terminal held at a temperature $T_P$, with which the electrons exchange energy. The nanostructure [central (gray) disk], placed in a magnetic field $B$ as in (a), is modeled by an Aharonov-Bohm ring threaded by a magnetic flux $\mathrm{\Phi }$ as in (b). The electrons exchange energy with vibrational modes on a dot placed on the upper arm.

Figure 2

The efficiency at maximal power of the E2T device (scaled by the Carnot efficiency ${\eta }_{\mathrm{rev}}$), Eq. (17), as a function of the asymmetry parameter $a$, for ${\zeta }_{\max }=10$; $\zeta =8$ [the dashed (red) curve], 9 [the dotted (green) curve], and 10 [the thick (blue) line]. The horizontal line is $\eta (V_{\mathrm{MP}},\zeta ={\zeta }_{\max })/{\eta }_{\mathrm{rev}}$ of the 2T junction.

Figure 3

The E2T figure of merit $\tilde{\zeta }$, Eq. (20) (scaled by the 2T one, $\zeta$), as a function of $a$, for ${\zeta }_{\max }=10$; $\zeta =8$ [the dashed (red) curve], 9 [the dotted (green) curve], 9.5 [the thick (blue) line], 9.9 [the thin (black) curve], and 9.95 [the thin dashed (black) curve]. Inset: The range where $\tilde{\zeta }/\zeta$ crosses 1.