Stochastically driven single-level quantum dot: A nanoscale finite-time thermodynamic machine and its various operational modes

Abstract

We describe a single-level quantum dot in contact with two leads as a nanoscale finite-time thermodynamic machine. The dot is driven by an external stochastic force that switches its energy between two values. In the isothermal regime, it can operate as a rechargeable battery by generating an electric current against the applied bias in response to the stochastic driving and then redelivering work in the reverse cycle. This behavior is reminiscent of the Parrondo paradox. If there is a thermal gradient the device can function as a work-generating thermal engine or as a refrigerator that extracts heat from the cold reservoir via the work input of the stochastic driving. The efficiency of the machine at maximum power output is investigated for each mode of operation, and universal features are identified. Parallel to spectacular developments in bio-and nanotechnology, there has been great theoretical interest in the study of small-scale machines. A well-documented case is the small-scale Carnot engine, in which the operational unit is subject to thermal fluctuations [1][2][3][4]. Of greater biological relevance are machines that convert one form of work to another, and yet these have received far less attention [5]. In this article we introduce an electronic nanodevice that allows several modes of operation. The device is a single-level quantum dot subject to stochastic driving while in contact with two reservoirs that may be at different temperatures and chemical potentials. Its properties can be derived from a stochastic thermodynamic description [6]. We investigate in analytic detail various operational regimes. When operating under tight coupling conditions, familiar features are recovered in appropriate limits: Carnot efficiency for reversible operation when the reservoirs are at different temperatures, universal features of efficiency at maximum power [3,7], and efficiency at maximum power close to the Curzon-Ahlborn efficiency [8]. When the reservoirs are at the same temperature, the work done on the dot by the switching can reverse the "normal" direction (from high to low chemical potential) of the current. Thus, the engine can be seen as a technologically relevant implementation of the Parrondo paradox [9] in that the switching can induce an electron flow against the chemical gradient. When operating under tight coupling, the efficiency at maximum power starts from the universal value of 1/2 close to equilibrium and increases monotonically to 1 as one moves further into the nonequilibrium regime and can thus be much higher than in the traditional implementations of the Parrondo paradox [10]. The same efficiency is observed when the engine works in the reverse mode. I. MODEL AND DYNAMICSWe consider a single-level quantum dot whose energy is stochastically switched between an upper and a lower value, ε j , with j = u or d (Fig. 1). The upward and downward rates are k + and k − . The dot is in contact with a left and a right lead, ν = L or R, at chemical potentials μ ν and temperatures T ν . The transition r...

Universal theory of efficiency fluctuations

“…The statistical properties of the efficiency provide a much more accurate characterization of the performance of small machines than the macroscopic efficiency. In view of the high interest in recent years for the study of finite-time thermodynamics at small scales [7,[35][36][37][38][39][40][41][42][43][44], we expect that the study of efficiency fluctuations will become a paradigm in this field. Finally, let us emphasize that the predictions of our theory for efficiency fluctuations provide a way to verify the implications of the fluctuation theorem, which can be seen as the generalization of the second law for small systems.…”

Mechanical Autonomous Stochastic Heat Engine

“…In the Supplemental Material [29] we present alternative designs that attain efficiencies up to 50% of the theoretical maximum. When the temperature ratio T H =T C is close to 1, the energy flow between T H and T C reverses, and the machine acts as a refrigerator [23]. The refrigerator regime requires a constant supply of energy to the cantilever.…”

“…A central problem in stochastic thermodynamics is the construction and analysis of stochastic heat engines, the low-dimensional analogs of conventional thermal machines. A stochastic heat engine is a low-dimensional device that operates between two thermal baths at different temperatures, and is able to produce work while suppressing the randomness inherent in thermal motion [1,[4][5][6][7][21][22][23]. Thermal engine operation is characterized by the presence of nonpassive states of motion, which have lower entropy (for the same energy) than equilibrium states [24,25] and therefore allow the extraction of energy without an associated entropy flow [1].…”

Relative thermalization

“…Today, it is also being applied to study heat and work flows in the micro and nano regimes. In fact, advances in the manipulation of small systems have allowed us to extract work from systems such as quantum dots and trapped ions [1,2]. Yet, thermodynamics as a science is still adapting to this new regime, and it still bears some of the traits of the gaseous systems for which it was first designed.…”

“…Under certain natural conditions, like weak coupling, π S approximates a Gibbs state [9]. 1 In order to better understand this definition, we note that, as knowledge is relative, so is thermalization. An observer who can only measure a few parameters of the system might see it as thermalized, while someone with more precise measurement instruments (like Maxwell's demon) may see a well-defined microstate.…”