Thermodynamic and quantum bounds on nonlinear dc thermoelectric transport

I consider the nonequilibrium dc transport of electrons through a quantum system with a thermoelectric response. This system may be any nanostructure or molecule modeled by the nonlinear scattering theory, which includes Hartree-like electrostatic interactions exactly, and certain dynamic interaction effects (decoherence and relaxation) phenomenologically. This theory is believed to be a reasonable model when single-electron charging effects are negligible. I derive three fundamental bounds for such quantum systems coupled to multiple macroscopic reservoirs, one of which may be superconducting. These bounds affect nonlinear heating (such as Joule heating), work and entropy production. Two bounds correspond to the first law and second law of thermodynamics in classical physics. The third bound is quantum (wavelength dependent), and is as important as the thermodynamic ones in limiting the capabilities of mesoscopic heat engines and refrigerators. The quantum bound also leads to Nernst's unattainability principle that the quantum system cannot cool a reservoir to absolute zero in a finite time, although it can get exponentially close.

Figure 1

Quantum systems with $K_{\mathrm{n}}=3$ normal reservoirs and one superconducting (sc) reservoir. Examples include (a) nanostructures such as those defined by electrostatic gates on top of a two-dimensional electron gas (2deg) or (b) molecules. Charge and heat currents, $I_i$ and $J_i$, are positive if they flow from reservoir $i$ into the quantum system.

Figure 2

A quantum system with a finite thermoelectric response being used as a heat engine providing power to a set of loads. The sc reservoir is superconducting, but “load sc” is not (its simply the load coupled to the sc reservoir).