Cooling by Heating: Refrigeration Powered by Photons

Abstract

In a recent letter, Cleuren et. al. [1] proposed a mechanism for solar refrigeration composed of two metallic leads mediated by two coupled quantum dots and powered by (solar) photons. In their analysis the refrigerator can operate to T r → 0 and the cooling flux ˙ Q r ∝ T r. We comment that this model violates the dynamical version of the III-law of thermodynamics. There are seemingly two independent formulation of the third law. The first, known as the Nernst heat theorem, implies that the entropy flow from any substance at absolute zero temperature is zero. At steady state the second law implies that the total entropy production is non-negative, i − ˙ Qi Ti ≥ 0 where ˙ Q i is positive for heat flowing into the system from the i-th bath. In order to insure the fulfillment of the second law when one of the heat baths (labeled k) approaches the absolute zero temperature. It is necessary that the entropy production from this bath scales as ˙ S k ∼ T α k with α ≥ 0. For the case where α = 0 the fulfillment of the second law depends on the entropy production of the other baths, which should compensate on the negative entropy production of the k bath. The first formulation of the third law slightly modifies this restriction. Instead of α ≥ 0 the third low impose α > 0 guaranteeing that at the absolute zero ˙ S k = 0. The second formulation of the third law is a dynamical one, known as the unatinability principle: No refrigerator can cool a system to absolute zero temperature at finite time. This formulation is more restrictive, imposing limitations on the spectral density and the dispersion dynamics of the heat bath [2]. We quantify this formulation by evaluating the characteristic exponent ζ of the cooling process dT (t) dt ∼ −T ζ , T → 0 (1) Namely for ζ < 1 the system is cooled to zero temperature at finite time. Eq.(1) can be related to the heat flow: ˙ Q k (T k (t)) = −c V (T k (t)) dT k (t) dt (2) where c V is the heat capacity of the bath. The refrigerator presented in [1] violates the III-law as in Eqs. (2) and (1). For an electron reservoir at low temperature the heat capacity c V ∼ T. The heat current of the refrigerator of [1] ˙ Q r ∝ T r therefore one obtains ζ = 0 hence zero temperature is achieved at finite time, in contradiction with the third law. Finding the flow in the analysis of [1] is not a trivial task. A possible explanation emerges from the assumption in [1] that transitions between lower and higher levels within the individual dots are negligible. Photon assisted tunneling between dots produce a week tunnel current [3]. In comparison quenching transitions in the individual dots cannot be neglected. A modified master equation which includes these transitions can be constructed for a five level system: ˙ p = M · p where p = (p 0 , p ld , p rd , p lu , p ru) T. Where p 0 is the probability of finding no electron in the double dot and p ij is the probability of finding one electron in the corresponding energy level, with l-left, r-right, d-down, u-up. The M matrix is 5x5 matrix which includes al...

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