Cooling by heating: Restoration of the third law of thermodynamics

We have made a simple and natural modification of a recent quantum refrigerator model presented by Cleuren et al. [Phys. Rev. Lett. 108, 120603 (2012)]. The original model consist of two metal leads acting as heat baths and a set of quantum dots that allow for electron transport between the baths. It was shown to violate the dynamic third law of thermodynamics (the unattainability principle, which states that cooling to absolute zero in finite time is impossible). By taking into consideration the finite energy level spacing Δ, in metals we restore the third law while keeping all of the original model's thermodynamic properties intact down to the limit of k(B)T ∼ Δ, where the cooling rate is quenched. The spacing Δ depends on the confinement of the electrons in the lead and therefore, according to our result larger samples (with smaller level spacing), could be cooled efficiently to lower absolute temperatures than smaller ones. However, a large lead makes the assumption of instant equilibration of electrons implausible; in reality one would only cool a small part of the sample and we would have a nonequilibrium situation. This property is expected to be model independent and raises the question whether we can find an optimal size for the lead that is to be cooled.