PRL 112, 048901 (2014)

Comment on “Cooling by Heating: Refrigeration Powered by Photons” Whereas absorption refrigeration that can use heat for cooling had already been known to work in the early twentieth century, its efficiency was low, had moving parts, and was noisy. Recently, Cleuren et al. presented a novel solid-state (mesoscopic) model for a refrigerator operating between two electronic baths held at two different temperatures [1], suggesting the Sun as a possibility for the third hot bath supplying the required energy. (A different optomechanical model was suggested in Ref. [2].) However, they found that from the assumptions they made on the model it followed that in the low-temperature regime the rate at which heat is pumped from the lower-temperature reservoir scales as that bath’s temperature. Two Comments [3,4], apparently acquiesced to by the authors [5], challenged the correctness of this spectacular suggestion. Both pointed out that it contradicts the “dynamic version” of the third law that states that no refrigerator can cool a system to absolute zero during a finite time. Here, we show that an elementary treatment of a simplified model removes the problem. The idea of Ref. [1] was fine, but a specific assumption on the parameters has caused the difficulty. We consider the two-level model of Ref. [6]; see Fig. 1 there. A boson source at a temperature T ph drives current between the left and the right electronic reservoirs through the levels E1 and E2 ð> E1 Þ on the two “quantum dots.” As in Ref. [1] we assume that the bias voltage vanishes, i.e., μL ¼ μR ≡ μ. The left reservoir is cooled by moving heat to the right one so that T L < T R . We first consider the linearresponse regime, where all three temperatures are close to the common temperature T of the system, which is taken to be low (the case when T ph is high will be discussed later). We assume that the boson-assisted hopping is the strongly dominant electronic transport channel [6]. Any relevant inelastic transmission will be exponentially small as the temperature approaches zero. As in previous publications [1,3,4], we discuss noninteracting quasiparticles. The main effect of the interactions is expected to be a renormalization of the model parameters [7,8], which is not strongly temperature dependent and, therefore, will not compensate for the exponential decay of the (inelastic) transition rates at low temperatures. An electron that exits the left electronic bath at energy E1 carries heat E1 − μ. To cool that bath, we must take E1 − μ > 0. In the calculation of the electric and heat currents emerging from the left bath by the golden rule [6], these currents are proportional to the population of E1 , which is exponentially small for kB T ≪ E1 − μ. Hence, the cooling power is exponentially small at low enough temperatures. An analogous condition holds for the two

0031-9007=14=112(4)=048901(1)

week ending 31 JANUARY 2014

PHYSICAL REVIEW LETTERS

levels [1] below μ. This resolves the serious problem stemming from the fact that the cooling rate was found to be proportional to T at low T, which contradicted the dynamic version of the third law [3,4]. That result followed from taking one level in the model to approach the chemical potential linearly with the temperature, which is unnecessary and complicates the setup. We could have invoked kB T ≪ E2 − E1 . Obviously, that energy difference is provided by the bosons coming from the thermal bath, necessitating kB T ph of the order of E2 − E1 . However, we gave the argument using E1 − μ, because this allows the boson reservoir to be very hot (e.g., 6000°C [1]) as long as the electrons are kept at their temperatures close to T. We can then have linear transport between the left and the right reservoirs, as long as the boson source does not heat them. Research supported by the Israeli Science Foundation (ISF) and the US-Israel Binational Science Foundation (BSF). Y.I. thanks J. von Delft for his hospitality in LMU Munich. We thank B. Landa, J-H. Jiang, and E. Shahmoon for instructive discussions. O. Entin-Wohlman1,2,3 and Y. Imry3 1

Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University Tel Aviv 69978, Israel 2 Department of Physics and the Ilse Katz Center for Meso- and Nano-Scale Science and Technology Ben Gurion University Beer Sheva 84105, Israel 3 Department of Condensed Matter Physics Weizmann Institute of Science Rehovot 76100, Israel Received 7 November 2013; published 30 January 2014 DOI: 10.1103/PhysRevLett.112.048901 PACS numbers: 05.70.Ln [1] B. Cleuren, B. Rutten, and C. Van den Broeck, Phys. Rev. Lett. 108, 120603 (2012). [2] A. Mari and J. Eisert, Phys. Rev. Lett. 108, 120602 (2012). [3] A. Levy, R. Alicki, and R. Kosloff, Phys. Rev. Lett. 109, 248901 (2012). [4] A. E. Allahverdyan, K. V. Hovhannisyan, and G. Mahler, Phys. Rev. Lett. 109, 248903 (2012). [5] B. Cleuren, B. Rutten, and C. Van den Broeck, Phys. Rev. Lett. 109, 248902 (2012). [6] J-H. Jiang, O. Entin-Wohlman, and Y. Imry, Phys. Rev. B 85, 075412 (2012). [7] M. Pollak, Discuss. Faraday Soc. 50, 13 (1970). [8] A. L. Efros and B. I. Shklovskii, J. Phys. C 8, L49 (1975).

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