From local force-flux relationships to internal dissipations and their impact on heat engine performance: The illustrative case of a thermoelectric generator
Yann Apertet, Henni Ouerdane, Christophe Goupil, Philippe Lecoeur
We present an in-depth analysis of the sometimes understated role of the principle of energy conservation in linear irreversible thermodynamics. Our case study is that of a thermoelectric generator (TEG), which is a heat engine of choice in irreversible thermodynamics, owing to the coupling between the electrical and heat fluxes. We show why Onsager's reciprocal relations must be considered locally and how internal dissipative processes emerge from the extension of these relations to a global scale: the linear behavior of a heat engine at the local scale is associated with a dissipation process that must partake in the global energy balance. We discuss the consequences of internal dissipations on the so-called efficiency at maximum power, in the light of our comparative analyses of exoreversibility and endoreversibility on the one hand, and of two classes of heat engines, autonomous and periodically-driven, on the other hand. Finally, basing our analysis on energy conservation, we also discuss recent works which claim the possibility to overcome the traditional boundaries on efficiency imposed by finite-time thermodynamics in thermoelectric systems with broken time-reversal symmetry; this we do by introducing a "thermal" thermopower and an "electrical" thermopower which permits an analysis of the thermoelectric response of the TEG considering a possible dissymmetry between the electrical/thermal and the thermal/electrical couplings.
Thermoelectric generators are particularly suitable to investigate the irreversible processes which govern the coupled transport of matter and heat in solid-state systems. We study the efficiency at maximum power in the strong coupling regime, where the thermal flux is proportionnal to the electrical current inside the generator. We demonstrate that depending on the source of irreversibility we obtain either the Curzon-Ahlborn efficiency for external dissipation or a universal efficiency at maximum power for internal dissipation. A continuous change between these two extremes is evidenced. Eff…
Improvement of thermoelectric systems in terms of performance and range of applications relies on progress in materials science and optimization of device operation. In this chapter, we focuse on optimization by taking into account the interaction of the system with its environment. For this purpose, we consider the illustrative case of a thermoelectric generator coupled to two temperature baths via heat exchangers characterized by a thermal resistance, and we analyze its working conditions. Our main message is that both electrical and thermal impedance matching conditions must be met for opti…
We propose a simple classical dynamical model of a thermoelectric (or thermochemical) heat engine based on a pair of ideal gas containers connected by two unequal scattering channels. The model is solved analytically and it is shown that a suitable combination of parameters can be chosen such that the engine operates at Carnot's efficiency.
We demonstrate the possibility to build a thermoelectric engine using a one dimensional gas of molecules with unequal masses and hard-point interaction. Most importantly, we show that the efficiency of this engine is determined by a new parameter YT which is different from the well known figure of merit ZT. Even though the efficiency of this particular model is low, our results shed new light on the problem and open the possibility to build efficient thermoelectric engines.
Theoretical thermoelectric nanophysics models of low-dimensional electronic heat engine and refrigerator devices, comprising two-dimensional hot and cold reservoirs and an interconnecting filtered electron transport mechanism have been established. The models were used to numerically simulate and evaluate the thermoelectric performance and energy conversion efficiencies of these low-dimensional devices, based on three different types of electron transport momentum-dependent filters, referred to herein as: kx, ky and kr filters. Assuming the Fermi-Dirac distribution of electrons, expressions fo…
Mesoscopic thermoelectric heat engine is much anticipated as a device that allows us to utilize with high efficiency wasted heat inaccessible by conventional heat engines. However, the derivation of the heat current in this engine seems to be either not general or described too briefly, even inappropriate in some cases. In this paper, we give a clear-cut derivation of the heat current of the engine with suitable assumptions beyond the linear-response regime. It resolves the confusion in the definition of the heat current in the linear-response regime. After verifying that we can construct the …
The quest to identify the best heat engine has been at the center of science and technology. Thermoelectric nanoscale heat engines convert heat flows into useful work in the form of electrical power and promise the realization of on-chip power production. Considerable studies have so far revealed the potentials to yield an enhanced efficiency originating from quantum confinement effects and energy-dependent transport properties. However, the full benefit of many-body interactions in thermoelectric is yet to be investigated; identifying the optimal interaction is a hard problem due to combinato…
We study the performance of a three-terminal thermoelectric device such as heat engine and refrigerator with broken time-reversal symmetry by applying the unified trade-off figure of merit ($\dotΩ$ criterion) which accounts for both useful energy and losses. For heat engine, we find that a thermoelectric device working under the maximum $\dotΩ$ criterion gives a significantly better performance than a device working at maximum power output. Within the framework of linear irreversible thermodynamics such a direct comparison is not possible for refrigerators, however, our study indicates that, f…
We analyze a steady-state thermoelectric engine, whose working substance consists of two capacitively coupled quantum dots. One dot is tunnel-coupled to a hot reservoir serving as a heat source, the other one to two electrically biased reservoirs at a colder temperature, such that work is extracted under the form of a steady-state current against the bias. In single realizations of the dynamics of this steady-state engine autonomous, 4-stroke cycles can be identified. The cycles are purely stochastic, in contrast to mechanical autonomous engines which exhibit self-oscillations. In particular, …
In the previous paper of Alicki et.al. a model of a solar cell has been proposed in which the non-periodic source of energy - photon flux - drives the collective periodic motion of electrons in a form of plasma oscillations. Subsequently, plasma oscillations are rectified by the p-n junction diode into the direct current (work). This approach makes a solar cell similar to standard macroscopic heat motors or turbines which always contain two heat baths, the working medium and the periodically moving piston or rotor. Here, a very similar model is proposed in order to describe the operation princ…
Power and efficiency of heat engines are two conflicting objectives, and a tight efficiency bound is expected to give insights on the fundamental properties of the power-efficiency tradeoff. Here we derive an upper bound on the efficiency of steady-state heat engines, which incorporates higher-order fluctuations of the power. In a prototypical model of nonlinear nanostructured thermoelectrics, we show that the obtained bound is tighter than a well-established efficiency bound based on the thermodynamic uncertainty relation, demonstrating that the higher-order terms have rich information about …
We identify the operational conditions for maximum power of a nanothermoelectric engine consisting of a single quantum level embedded between two leads at different temperatures and chemical potentials. The corresponding thermodynamic efficiency agrees with the Curzon-Ahlborn expression up to quadratic terms in the gradients, supporting the thesis of universality beyond linear response.
Thermoelectric transport involving an arbitrary number of terminals is discussed in the presence of a magnetic field breaking time-reversal symmetry within the linear response regime using the Landauer-Büttiker formalism. We derive a universal bound on the Onsager coefficients that depends only on the number of terminals. This bound implies bounds on the efficiency and on efficiency at maximum power for heat engines and refrigerators. For isothermal engines pumping particles and for absorption refrigerators these bounds become independent even of the number of terminals. On a technical level, …
While Carnot's model engines demonstrate ideal performances regarding conversion efficiency, they cannot be actually used as energy converters since they are non causal systems. Such an unphysical behavior indeed restrains the working conditions to a single point where, in the case of a refrigerator (generator), the cooling power (output power) vanishes. Focusing on the example of a thermoelectric refrigerator, we study the impact of different dissipation sources on the causality of such systems. Basing our analysis on the block diagram description of this system, we discuss particularly the f…
Recently, a thermodynamic uncertainty relation (TUR) has been formulated for classical Markovian systems demonstrating trade-off between precision (current fluctuation) and cost (dissipation). Systems that violate the TUR are interesting as they overcome another trade-off relation concerning the efficiency of a heat engine, its power, and its stability (power fluctuations). Here, we analyze the root, extent, and impact on performance of TUR violations in quantum thermoelectric junctions at steady state. Considering noninteracting electrons, first we show that only the "classical" component of …
Thermoelectric devices are heat engines, which operate as generators or refrigerators using the conduction electrons as a working fluid. The thermoelectric heat-to-work conversion efficiency has always been typically quite low, but much effort continues to be devoted to the design of new materials boasting improved transport properties that would make them of the electron crystal-phonon glass type of systems. On the other hand, there are comparatively few studies where a proper thermodynamic treatment of the electronic working fluid is proposed. The present article aims to contribute to bridge…
We show how the formalism used for thermoelectric transport may be adapted to Smoluchowski's seminal thought experiment also known as Feynman's ratchet and pawl system. Our analysis rests on the notion of useful flux, which for a thermoelectric system is the electrical current and for Feynman's ratchet is the effective jump frequency. Our approach yields original insight into the derivation and analysis of the system's properties. In particular we define an entropy per tooth in analogy with the entropy per carrier or Seebeck coefficient, and we derive the analogue to Kelvin's second relation f…
The study shows that presence of the quantum coherent, unitary component of the evolution of the system can improve constancy of heat engines, i.e., decrease fluctuations of the output power, in comparison with purely stochastic setups. This enables to overcome the recently derived trade-off between efficiency, power and constancy, which applies to classical Markovian steady-state heat engines. The concept is demonstrated using a model system consisting of two tunnel-coupled orbitals (i.e., electronic levels), each attached to a separate electronic reservoir; such a setup can be realized, for …
In this paper we revisit the thermocouple model, as a linear irreversible thermodynamic energy converter. As is well known, the linear model of the thermocuple is one of the classics in this branch. In this model we note two types of phenomenological coefficients: the first comes from some microscopic models, such as the coefficient associated with the electric conductivity, and the second comes from experimental facts such as the coefficient associated with the thermoelectric power. We show that in the last case, these coefficients can be related to the operation modes of the converter. These…
The design of a mesoscopic self-oscillating heat engine that works thanks to purely quantum effects is presented. The proposed scheme is amenable to experimental implementation with current state-of-the-art nanotechnology and materials. One of the main features of the structure is its versatility: The engine can deliver work to a generic load without galvanic contact. This makes it a promising building block for low-temperature on-chip energy management applications. The heat engine consists of a circuit featuring a thermoelectric element based on a ferromagnetic insulator-superconductor tunne…
Steady-state thermoelectric machines convert heat into work by driving a thermally-generated charge current against a voltage gradient. In this work, we propose a new class of steady-state heat engines operating in the quantum regime, where a quasi-periodic tight-binding model that features a mobility edge forms the working medium. In particular, we focus on a generalization of the paradigmatic Aubrey-André-Harper (AAH) model, known to display a single-particle mobility edge that separates the energy spectrum into regions of completely delocalized and localized eigenstates. Remarkably, these t…
