is just a way to characterize the efficiency, via $$\begin{aligned}
\eta_{\rm eng} = \eta_{\rm eng}^{\rm carnot}  {\sqrt{ZT+1} -1 \over \sqrt{ZT+1} +1}, 
\nonumber\end{aligned}$$ with a similar relationship for refrigerators. Thus, someone asking for a device with a $ZT > 3$, actually requires one with an efficiency of more than one third of Carnot efficiency. This is crucial, because the efficiency is a physical quantity in linear and nonlinear situations, while $ZT$ is only meaningful in the linear-response regime [@Zebarjadi2007; @Grifoni2011; @2012w-pointcont; @Meair-Jacquod2013; @Michelini2014; @Azema-Lombardo-Dare2014].

Linear-response theory rarely fails for bulk semiconductors, even when $T_L$ and $T_R$ are very different. Yet it is completely [*inadequate*]{} for the quantum systems that we consider here. Linear-response theory requires the temperature drop on the scale of the electron relaxation length $l_{\rm rel}$ (distance travelled before thermalizing) to be much less than the average temperature. For a typical millimetre-thick bulk thermoelectric between a diesel motor’s exhaust system ($T_L\simeq 700$K) and its surroundings ($T_R\simeq 280$K), the relaxation length (inelastic scattering length) is of order the mean free path; typically 1-100nm. The temperature drop on this scale is tens of thousands of times smaller than the temperature drop across the whole thermoelectric. This is absolutely tiny compared with 