Theory of single-electron heat engines coupled to electromagnetic environments

We introduce a new class of mesoscopic heat engines consisting of a tunnel junction coupled to a linear thermal bath. Work is produced by transporting electrons up against a voltage bias like in ordinary thermoelectrics but heat is transferred by microwave photons, allowing the heat bath to be widely separated from the electron system. A simple and generic formalism capable of treating a variety of different types of junctions and environments is presented. We identify the systems and conditions required for maximal efficiency and maximal power. High efficiencies are possible with quantum dot arrays but high power can be achieved also with metallic systems.

Figure 1

(a) Schematic of a single-electron heat engine coupled to an external environment with impedance $Z\left(\omega \right)$. Electrons tunneling through the left junction exchange energy with the heat bath, enabling net current against the voltage bias. (b), (c) Energy-level diagrams of junction systems with one or two dots between the leads. One of the junctions, marked with a photon symbol, is coupled to the external bath. The photon-assisted tunneling rates are ${\Gamma }_{\pm }$.

Figure 2

Maximum power production for (a) a junction between two quantum dots, and (b) a junction between a quantum dot and a metal. (Top) The maximum power for given temperatures. Note the different units of power for (a) and (b). (Middle) efficiency at maximum power. (Bottom) values of $V$ (full line), $E$ (dashed), and $E_C$ (dash-dotted) that give the maximum power. The temperature difference is $\Delta T=T_{\mathrm{env}}-T$ and the average temperature is $T_0=\frac{1}{2}(T+T_{\mathrm{env}})$.