whole loop is squashed onto the $P_{\rm gen}=0$ axis. In linear-response language, this machine has $ZT \to \infty$. In this limit, the efficiency at maximum power can be very close to that of Curzon and Ahlborn [@Esposito2009-thermoelec] (the star in Fig. \[Fig:summary\]), just as its maximum efficiency can be that of Carnot[@Humphrey-Linke2005] (see previous section). However, its maximum power output is $\propto { e^{\operatorname{-}} }V\Delta/\hbar$ for small $\Delta$ (where $V$ is finite, chosen to ensure maximum power), which vanishes for $\Delta \to 0$, although it is much larger than Eq. (\[Eq:power-for-Delta-to-zero\]). Fig. \[Fig:summary\] shows that a system with larger $\Delta$ (such as the red curve) operating near its maximum efficiency will have both higher efficiency and higher power output than the one with small $\Delta$ (left most grey curve) operating at maximum power.

This article shows how to derive the thick black curve in Fig. \[Fig:summary\], thereby showing that there is a fundamental trade-off between efficiency and power output in optimal thermodynamic machines made from thermoelectrics [@footnote:casati-review]. As such, our work overturns the idea that maximizing efficiency at maximum power is the best route to machines with both high efficiency and high power. It also overturns the idea that systems 