We study the thermodynamic efficiency of a nanosized photoelectric device and show that at maximum power output, the efficiency is bounded from above by a result closely related to the Curzon-Ahlborn efficiency. We find that this upper bound can be attained in nanosized devices displaying strong coupling between the generated electron flux and the incoming photon flux from the sun. DOI: 10.1103/PhysRevB.80.235122 PACS number͑s͒: 05.70.Ln, 05.30.Ϫd, 73.50.Pz Understanding and controlling the mechanisms that determine the efficiency of photoelectric devices is of fundamental importance in the quest for efficient and clean sources of energy. Thermodynamically speaking, these devices are driven by the temperature difference between a hot reservoir ͑sun, temperature T s ͒ and a cold reservoir ͑earth, ambient temperature T͒. Therefore, like any heat engine, the efficiency at which the conversion of radiation into electrical energy takes place has a universal upper bound given by the Carnot efficiency ͑1͒Although this result has fundamental theoretical implications, it is of poor practical use since it is only reached when the device is operating under reversible conditions. Hence the generated power, defined as the output energy divided by the ͑infinite͒ operation time, goes to zero. In realistic circumstances of finite power output, the efficiency will necessarily be below the Carnot limit due to irreversible processes taking place in the device. Another source of possible efficiency decrease are energy losses within the device for example due to nonradiative recombination of charge carriers. Since the operational parameters of the device are mostly determined in such a way that a maximum power output is obtained, Curzon and Ahlborn examined in 1975 the efficiency of a Carnot cycle with a finite cycling time and, using the endoreversible approximation, found an efficiency at maximum2 This result is remarkable since it does not depend on the specific details of the system, and thus the question of universality naturally arises. Recent works [3][4][5][6][7] have indeed demonstrated that in the linear regime ͑small temperature differences, c Ӷ 1͒ the Curzon-Ahlborn efficiency is universal for so-called strongly coupled systems, where the heat and work producing fluxes are proportional. In these systems internal energy losses are absent, implying that the resulting efficiency is exclusively determined by the ͑unavoidable͒ irreversible processes occurring at finite power. Hence, at least in the linear regime, the CurzonAhlborn efficiency is indeed a universal upper bound, with a similar status as the Carnot efficiency. In the nonlinear regime, the efficiency at maximum power becomes device dependent but is again found to be highest for strongly coupled systems. Remarkably, it remains closely related to the Curzon-Ahlborn result. 6,7 While energy losses are almost unavoidable in the macroscopic world, new technological developments at the nanoscale open up the road to highly efficient devices. In thermoelectric researc...

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...

We investigate the efficiency of power generation by thermochemical engines. For strong coupling between the particle and heat flows and in the presence of a left-right symmetry in the system, we demonstrate that the efficiency at maximum power displays universality up to quadratic order in the deviation from equilibrium. A maser model is presented to illustrate our argument. The concept of Carnot efficiency is a cornerstone of thermodynamics. It states that the efficiency of a cyclic (''Carnot'') thermal engine that transforms an amount Q ðrÞ of energy extracted from a heat reservoir at temperature T r into an amount of work W is at most ¼ W =Q ðrÞ c ¼ 1 À T l =T r , where T l is the temperature of a second, colder reservoir. The theoretical implications of this result are momentous, as they lie at the basis of the introduction by Clausius of the entropy as a state function. The practical implications are more limited, since the upper limit c (''Carnot efficiency'') is only reached for engines that operate reversibly. As a result, when the efficiency is maximal, the output power is zero. By optimizing the Carnot cycle with respect to power rather than efficiency, Curzon and Ahlborn found that the corresponding efficiency is given by. They obtained this result for a specific model, using in addition the socalled endo-reversible approximation (i.e., neglecting the dissipation in the auxiliary, work producing entity). Subsequently, the validity of this result as an upper bound, as well as its universal character, were the subjects of a long-standing debate. In the regime of linear response, more precisely to linear order in c , it was proven that the efficiency at maximum power is indeed limited by the Curzon-Ahlborn efficiency, which in this regime is exactly half of the Carnot efficiency,. The upper limit is reached for a specific class of models, namely, those for which the heat flux is strongly coupled (i.e., directly proportional) to the work-generating flux. Interestingly, such strong coupling is also a prerequisite for open systems to achieve Carnot efficiency [3,4]. In the nonlinear regime, no general result is known. Efficiencies at maximum power, not only below but also above Curzon-Ahlborn efficiency, have been reported [5][6][7][8]. However, it was also found, again in several strong coupling steady-state models [8][9][10] as well as for an overdamped Brownian particle in a time-dependent harmonic potential [7], that the efficiency at maximum power agrees with CA up to quadratic order in c , i.e., ¼c Þ, again raising the question of universality at least to this order. In this Letter we prove that the coefficient 1=8 is indeed universal for strong coupling models that possess a left-right symmetry. Such a universality is remarkable in view of the fact that most explicit macroscopic relationships, for example, the symmetry of Onsager coefficients, are limited to the regime of linear response. The interest in strong coupling is further motivated by the observation that it can naturally be achieved in nan...

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