Most Efficient Quantum Thermoelectric at Finite Power Output

Machines are only Carnot efficient if they are reversible, but then their power output is vanishingly small. Here we ask, what is the maximum efficiency of an irreversible device with finite power output? We use a nonlinear scattering theory to answer this question for thermoelectric quantum systems, heat engines or refrigerators consisting of nanostructures or molecules that exhibit a Peltier effect. We find that quantum mechanics places an upper bound on both power output and on the efficiency at any finite power. The upper bound on efficiency equals Carnot efficiency at zero power output but decays with increasing power output. It is intrinsically quantum (wavelength dependent), unlike Carnot efficiency. This maximum efficiency occurs when the system lets through all particles in a certain energy window, but none at other energies. A physical implementation of this is discussed, as is the suppression of efficiency by a phonon heat flow.

Figure 1

Typical thermoelectric devices are shown in (a) and (b), with (c) showing the quantum system with the thermoelectric response. To unify the analysis, heat is always taken to flow as shown; hence, a heat engine has $T_L>T_R$, while a refrigerator has $T_L<T_R$. In (a) and (b), the filled (open) circles are quantum systems where transport occurs via electron states above the Fermi surface (hole states below the Fermi surface). In (c), $N$ is the number of transverse modes in the narrowest part of the quantum system.

Figure 2

(a) Sketch of how the optimal transmission changes as the required power output is increased. Maximum power is when the right-hand edge of the boxcar function goes to $+\infty$. The qualitative features always follow this sketch, while the quantitative details depend on $T_R/T_L$. (b) Plots of optimal $\mathrm{\Delta }$ (solid curve) and $e^{-}V$ (dashed curve) against heat-engine power output $P_{\text{gen}}$ at $T_R/T_L=0.1$. The first equation in Eq. (12) then gives ${\epsilon }_0$. (c) Plots of optimal $\mathrm{\Delta }$ (solid curve) and $e^{-}V$ (dashed curve) against cooling power $J_L$ at $T_R/T_L=10$. The second equation in Eq. (14) then gives ${\epsilon }_1$.

Figure 3

Finding the ${\mathcal{T}}_{RL}^{\mu \mu }(\epsilon )$ function that maximizes the efficiency. In (a) ${\mathcal{T}}_{RL}^{\mu \mu }(\epsilon )$ is considered as infinitely many slices of width $\delta \to 0$, so slice $\gamma$ has energy ${\epsilon }_{\gamma }\equiv \gamma \delta$ and height ${\tau }_{\gamma }^{\mu }$. This gives (b) with a transcendental equation for ${\epsilon }_0$ and ${\epsilon }_1$.

Figure 4

Efficiencies of (a) heat engines and (b) refrigerators. The black lines are the maximum allowed efficiency for various power outputs as functions of $T_R/T_L$. The light green regions indicate allowed efficiencies for those power outputs.

Figure 5

Chain of systems with levels at energy $E_0$ and hoppings $\{t_i\}$ that will hybridize to form a band centered at $E_0$, with a width given by the hopping. If the hoppings are smallest at the chain’s middle and larger at its ends [50], the transmission becomes increasingly like a boxcar function as one increases $k$ (the bandwidths in the plots have been set to 1).