Comments on "Cooling by Heating: Refrigerator Powered by Photons

Comments on ”cooling by heating: Refrigeration powered by Photons” (1) arXiv:1208.2600v1 [quant-ph] 13 Aug 2012 (2) Amikam Levy(1) , Robert Alicki(2)(3) and Ronnie Kosloff(1) Institute of Chemistry The Hebrew University, Jerusalem 91904, Israel Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Poland and (3) Weston Visiting Professor, Weizmann Institute of Science, Rehovot, Israel 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 Tr → 0 and the cooling flux Q̇r ∝ Tr . 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 P i implies that the total entropy production is non-negative, i − Q̇ Ti ≥ 0 where Q̇i 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 Ṡk ∼ Tkα 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 Ṡk = 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) ∼ −T ζ , dt 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: Q̇k (Tk (t)) = −cV (Tk (t)) dTk (t) dt (2) where cV 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 cV ∼ T . The heat current of the refrigerator of [1] Q̇r ∝ Tr 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~ = (p0 , pld , prd , plu , pru )T . Where p0 is the probability of finding no electron in the double dot and pij 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 also quenching transition in the individual dots (see appendix). A crucial condition for the device to operate as a refrigerator [1] is that there is no net electric charging of the baths. Otherwise the electric current flowing through the device must be compensated by an external flow of electrons from the hot to the cold bath which would annihilate the cooling effect. When assuming that the relaxation rates within the individual dots are equal and finite, one obtains that the two manifolds of cooling and no net charging do not intersect in parameter space. As a result the condition for cooling and no net charging in the baths cannot be satisfied simultaneously. In conclusion, transitions in the individual dots, which are always present in real system, can not be neglected when treating electron transport in the double dot. The fulfillment of thermodynamical laws are a strong tool for verifying the quantum description of such nano-devices. For quantum description of refrigerator powered by heat (absorption refrigerator) see [2, 4]. 2 APPENDIX The M matrix reads         M =       ǫ +ǫ ǫ −ǫ ǫ −ǫ ǫ2 ǫ1 g g 1 2 + kb→s ) −(kb→s + kb→s + kb→s ǫ1 −ǫg kb→s g 1 ks→b ǫ1 −ǫg −ks→b − ǫ k↑g − ǫ +ǫ ǫ1 ks→b g 2 ks→b ǫ k↓g k↓∆l 0 k↓∆r k↑∆l  ǫ2 ks→b ǫ1 kb→s k↑g ǫ ǫ1 −ks→b − k↓g − k↑∆r 0 g 2 kb→s ǫ +ǫ k↑∆l 0 g 2 −ks→b − k↓g − k↓∆l k↑g ǫ2 kb→s 0 k↑∆r k↓g ǫ ǫ2 −ks→b − k↑g − k↓∆r ǫ ǫ +ǫ ǫ ǫ ǫ The energy difference between the lower and upper levels in the left and right dots are ∆l = ǫ2 − ǫ1 + 2ǫg and ∆r = ǫ2 − ǫ1 respectively. The transition rate between the bath and system is given by: ǫ kb→s = Γf (ǫ) ǫ ks→b = Γ(1 − f (ǫ)) (3) f (ǫ) = [exp((ǫ − µ)/T ) + 1]−1 The transition rates between dots are given by: k↑ǫ = Γs n(ǫ) = Γs (1 + n(ǫ)) n(ǫ) = [exp(ǫ/Ts ) − 1]−1 k↓ǫ ǫ (4) ǫ ∆ ∆ In the high temperature limit k↓g ≃ k↑g . For simplicity we assume that k↓∆r = k↑∆r = k↓ l = k↑ l ≡ k, see Fig 1 . We define Jrd (Jru ) as the particle current from the right bath into the ǫ1 (ǫ2 ) level and Jld (Jlu ) as the particle current from the left bath into the ǫ1 − ǫg (ǫ2 + ǫg ) level, respectively. ǫ1 ǫ1 Jrd = kb→s p0 − ks→b prd ǫ2 ǫ2 Jru = kb→s p0 − ks→b pru ǫ −ǫ ǫ −ǫ ǫ +ǫ ǫ +ǫ (5) g g 1 1 pld p0 − ks→b Jld = kb→s g g 2 2 Jlu = kb→s p0 − ks→b plu 000000000 000000000 000000000 000000000 kb↔s ε2+εg Tl 000000000 000000000 000000000 000000000 000000000 k µ000000000 000000000 000000000 000000000 000000000 000000000 000000000k ε1−εg 000000000 000000000 b↔s 000000000 kεg kεg 000000000 000000000 000000000 000000000 Tr 000000000 000000000 kb↔s 000000000 000000000 000000000 000000000 µ 000000000 000000000 000000000 000000000 kb↔s 000000000 000000000 000000000 000000000 000000000 ε2 k ε1 FIG. 1: Schematic representation of all possible electron transitions Negative current imply transition of particle from the system into the bath. The heat currents are given by: Q̇r = (ǫ1 − µ)Jrd + (ǫ2 − µ)Jru Q̇l = (ǫ1 − ǫg − µ)Jld + (ǫ2 + ǫg − µ)Jlu (6) As in [1] we take ǫ2 = −ǫ1 = ǫ and µ = 0. The condition of no net charging is Jrd + Jru = Jld + Jlu = 0 which eventually ǫ+ǫ reduces to Tǫr = T g (note that for ǫg = 0 no net charging is translated to Tl = Tr ). Calculating the heat flows under this l condition leads to: Q̇r = − Q̇l = − 4kΓǫsinh( ǫ+ǫg Tl ) ǫ+ǫg Tl ) ǫ+ǫg Tl ) 10k + Γ + (10k + 4Γ)cosh( ǫ+ǫ 4kΓ(ǫ + ǫg )sinh( T g ) l 10k + Γ + (10k + 4Γ)cosh( (7)               3 Both currents are negative implying that heat will always flow from the photon source into the baths. Therefor no refrigeration occurs. [1] B. Cleuren, B. Rutten, and C. Van den Broeck, Phys. Rev. Lett. 108, 120603 (2012). [2] A. Levy and R. Kosloff, Phys. Rev. Lett. 108, 070604 (2012). [3] W. G. van der Wiel, S. De Franceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Rev. Mod. Phys. 75, 1 (2002). [4] A. Levy, R. Alicki, and R. Kosloff, Phys. Rev. E 85, 061126 (2012).